Learning Objectives

Decompose (dfrac{P(x)}{Q(x)}), where

- (Q(x)) has only nonrepeated linear factors.
- (Q(x)) has repeated linear factors.
- (Q(x)) has a nonrepeated irreducible quadratic factor.
- (Q(x)) has a repeated irreducible quadratic factor.

Earlier in this chapter, we studied systems of two equations in two variables, systems of three equations in three variables, and nonlinear systems. Here we introduce another way that systems of equations can be utilized—the decomposition of rational expressions. Fractions can be complicated; adding a variable in the denominator makes them even more so. The methods studied in this section will help simplify the concept of a rational expression.

## Decomposing (frac{P(x)}{Q(x)}) where (Q(x)) Has Only Nonrepeated Linear Factors

Recall the algebra regarding adding and subtracting rational expressions. These operations depend on finding a common denominator so that we can write the sum or difference as a single, simplified rational expression. In this section, we will look at **partial fraction decomposition**, which is the undoing of the procedure to add or subtract rational expressions. In other words, it is a return from the single simplified **rational expression** to the original expressions, called the **partial fractions**.

For example, suppose we add the following fractions:

[dfrac{2}{x−3}+dfrac{−1}{x+2} onumber]

We would first need to find a common denominator: ((x+2)(x−3)).

Next, we would write each expression with this common denominator and find the sum of the terms.

[egin{align*} dfrac{2}{x-3}left(dfrac{x+2}{x+2} ight)+dfrac{-1}{x+2}left(dfrac{x-3}{x-3} ight)&= dfrac{2x+4-x+3}{(x+2)(x-3)}[4pt] &= dfrac{x+7}{x^2-x-6} end{align*}]

Partial fraction **decomposition** is the reverse of this procedure. We would start with the solution and rewrite (decompose) it as the sum of two fractions.

[ underbrace{dfrac{x+7}{x^2-x-6}}_{ ext{Simplified sum}} = underbrace{dfrac{2}{x-3}+dfrac{-1}{x+2}}_{ ext{Partial fraction decomposition }} onumber]

We will investigate rational expressions with linear factors and quadratic factors in the denominator where the degree of the numerator is less than the degree of the denominator. Regardless of the type of expression we are decomposing, the first and most important thing to do is factor the denominator.

When the denominator of the simplified expression contains distinct linear factors, it is likely that each of the original rational expressions, which were added or subtracted, had one of the linear factors as the denominator. In other words, using the example above, the factors of (x^2−x−6) are ((x−3)(x+2)), the denominators of the decomposed rational expression. So we will rewrite the simplified form as the sum of individual fractions and use a variable for each numerator. Then, we will solve for each numerator using one of several methods available for partial fraction decomposition.

PARTIAL FRACTION DECOMPOSITION OF (frac{P(x)}{Q(x)}): (Q(x)) HAS NONREPEATED LINEAR FACTORS

The *partial fraction decomposition* of (dfrac{P(x)}{Q(x)}) when (Q(x)) has nonrepeated linear factors and the degree of (P(x)) is less than the degree of (Q(x)) is

[dfrac{P(x)}{Q(x)}=dfrac{A_1}{(a_1x+b_1)}+dfrac{A_2}{(a_2x+b_2)}+dfrac{A_3}{(a_3x+b_3)}+⋅⋅⋅+dfrac{A_n}{(a_nx+b_n)}]

How to: Given a rational expression with distinct linear factors in the denominator, decompose it

- Use a variable for the original numerators, usually (A), (B), or (C), depending on the number of factors, placing each variable over a single factor. For the purpose of this definition, we use (A_n) for each numerator
(dfrac{P(x)}{Q(x)}=dfrac{A_1}{(a_1x+b_1)}+dfrac{A_2}{(a_2x+b_2)}+dfrac{A_3}{(a_3x+b_3)}+⋅⋅⋅+dfrac{A_n}{(a_nx+b_n)})

- Multiply both sides of the equation by the common denominator to eliminate fractions.
- Expand the right side of the equation and collect like terms.
- Set coefficients of like terms from the left side of the equation equal to those on the right side to create a system of equations to solve for the numerators.

Example (PageIndex{1}): Decomposing a Rational Function with Distinct Linear Factors

Decompose the given rational expression with distinct linear factors.

(dfrac{3x}{(x+2)(x−1)})

**Solution**

We will separate the denominator factors and give each numerator a symbolic label, like (A), (B) ,or (C).

(dfrac{3x}{(x+2)(x−1)}=dfrac{A}{(x+2)}+dfrac{B}{(x−1)})

Multiply both sides of the equation by the common denominator to eliminate the fractions:

((x+2)(x−1)left[ dfrac{3x}{(x+2)(x−1)} ight]=(x+2)(x−1)left[dfrac{A}{(x+2)} ight]+(x+2)(x−1)left[dfrac{B}{(x−1)} ight])

The resulting equation is

(3x=A(x−1)+B(x+2))

Expand the right side of the equation and collect like terms.

[egin{align*} 3x&= Ax-A+Bx+2B[4pt] 3x&= (A+B)x-A+2B end{align*}]

Set up a system of equations associating corresponding coefficients.

[egin{align*} 3&= A+B[4pt] 0&= -A+2B end{align*}]

Add the two equations and solve for (B).

[egin{align*} 3&= A+B[4pt] underline{0}&= underline{-A+2B}[4pt] 3&= 0+3B[4pt] 1&= B end{align*}]

Substitute (B=1) into one of the original equations in the system.

[egin{align*} 3&= A+1[4pt] 2&= A end{align*}]

Thus, the partial fraction decomposition is

(dfrac{3x}{(x+2)(x−1)}=dfrac{2}{(x+2)}+dfrac{1}{(x−1)})

Another method to use to solve for (A) or (B) is by considering the equation that resulted from eliminating the fractions and substituting a value for (x) that will make either the (A-) or (B-)term equal 0. If we let (x=1), the

(A-) term becomes 0 and we can simply solve for (B).

[egin{align*} 3x&= A(x-1)+B(x+2)[4pt] 3(1)&= A[(1)-1]+B[(1)+2][4pt] 3&= 0+3B[4pt] 1&= B end{align*}]

Next, either substitute (B=1) into the equation and solve for (A), or make the (B-)term (0) by substituting (x=−2) into the equation.

[egin{align*} 3x&= A(x-1)+B(x+2)[4pt] 3(-2)&= A[(-2)-1]+B[(-2)+2][4pt] -6&= -3A+0[4pt] dfrac{-6}{-3}&= A[4pt] 2&=A end{align*}]

We obtain the same values for (A) and (B) using either method, so the decompositions are the same using either method.

(dfrac{3x}{(x+2)(x−1)}=dfrac{2}{(x+2)}+dfrac{1}{(x−1)})

Although this method is not seen very often in textbooks, we present it here as an alternative that may make some partial fraction decompositions easier. It is known as the **Heaviside method**, named after Charles Heaviside, a pioneer in the study of electronics.

Exercise (PageIndex{1})

Find the partial fraction decomposition of the following expression.

(dfrac{x}{(x−3)(x−2)})

- Answer
(dfrac{3}{x−3}−dfrac{2}{x−2})

## Decomposing (frac{P(x)}{Q(x)}) where (Q(x)) Has Repeated Linear Factors

Some fractions we may come across are special cases that we can decompose into partial fractions with repeated linear factors. We must remember that we account for repeated factors by writing each factor in increasing powers.

PARTIAL FRACTION DECOMPOSITION OF (frac{P(x)}{Q(x)}): (Q(x)) HAS REPEATED LINEAR FACTORS

The partial fraction decomposition of (dfrac{P(x)}{Q(x)}), when (Q(x)) has a repeated linear factor occurringn n times and the degree of (P(x)) is less than the degree of (Q(x)), is

[dfrac{P(x)}{Q(x)}=dfrac{A_1}{(a_1x+b_1)}+dfrac{A_2}{(a_2x+b_2)}+dfrac{A_3}{(a_3x+b_3)}+⋅⋅⋅+dfrac{A_n}{(a_nx+b_n)}]

Write the denominator powers in increasing order.

How to: decompose a rational expression with repeated linear factors

- Use a variable like (A), (B), or (C) for the numerators and account for increasing powers of the denominators.[dfrac{P(x)}{Q(x)}=dfrac{A_1}{(a_1x+b_1)}+dfrac{A_2}{(a_2x+b_2)}+dfrac{A_3}{(a_3x+b_3)}+⋅⋅⋅+dfrac{A_n}{(a_nx+b_n)}]
- Multiply both sides of the equation by the common denominator to eliminate fractions.
- Expand the right side of the equation and collect like terms.
- Set coefficients of like terms from the left side of the equation equal to those on the right side to create a system of equations to solve for the numerators.

Example (PageIndex{2}): Decomposing with Repeated Linear Factors

Decompose the given rational expression with repeated linear factors.

(dfrac{−x^2+2x+4}{x^3−4x^2+4x})

**Solution**

The denominator factors are (x{(x−2)}^2). To allow for the repeated factor of ((x−2)), the decomposition will include three denominators: (x), ((x−2)), and ({(x−2)}^2). Thus,

(dfrac{−x^2+2x+4}{x^3−4x^2+4x}=dfrac{A}{x}+dfrac{B}{(x−2)}+dfrac{C}{{(x−2)}^2})

Next, we multiply both sides by the common denominator.

[egin{align*} x{(x-2)}^2left[ dfrac{-x^2+2x+4x}{{(x-2)}^2} ight]&= left[ dfrac{A}{x}+dfrac{B}{(x-2)}+dfrac{C}{{(x-2)}^2} ight]x{(x-2)}^2[4pt] -x^2+2x+4&= A{(x-2)}^2+Bx(x-2)+Cx end{align*}]

On the right side of the equation, we expand and collect like terms.

[egin{align*} -x^2+2x+4&= A(x^2-4x+4)+B(x^2-2x)+Cx[4pt] &= Ax^2-4Ax+4A+Bx^2-2Bx+Cx[4pt] &= (A+B)x^2+(-4A-2B+C)x+4A end{align*}]

Next, we compare the coefficients of both sides. This will give the system of equations in three variables:

Solving for (A) in Equation ef{2.3}, we have

[egin{align*} 4A&= 4[4pt] A&= 1 end{align*}]

Substitute (A=1) into Equation ef{2.1}.

Then, to solve for (C), substitute the values for (A) and (B) into Equation ef{2.2}.

[egin{align*} -4A-2B+C&= 2[4pt] -4(1)-2(-2)+C&= 2[4pt] -4+4+C&= 2[4pt] C&= 2 end{align*}]

Thus,

(dfrac{−x^2+2x+4}{x^3−4x^2+4x}=dfrac{1}{x}−dfrac{2}{(x−2)}+dfrac{2}{{(x−2)}^2})

Exercise (PageIndex{2})

Find the partial fraction decomposition of the expression with repeated linear factors.

(dfrac{6x−11}{{(x−1)}^2})

- Answer
[dfrac{6}{x−1}−dfrac{5}{{(x−1)}^2} onumber]

## Decomposing (frac{P(x)}{Q(x)}), where (Q(x)) Has a Nonrepeated Irreducible Quadratic Factor

So far, we have performed partial fraction decomposition with expressions that have had linear factors in the denominator, and we applied numerators (A), (B), or (C) representing constants. Now we will look at an example where one of the factors in the denominator is a quadratic expression that does not factor. This is referred to as an irreducible quadratic factor. In cases like this, we use a linear numerator such as (Ax+B), (Bx+C), etc.

DECOMPOSITION OF (frac{P(x)}{Q(x)}): (Q(x)) HAS A NONREPEATED IRREDUCIBLE QUADRATIC FACTOR

The partial fraction decomposition of (dfrac{P(x)}{Q(x)}) such that (Q(x)) has a nonrepeated irreducible quadratic factor and the degree of (P(x)) is less than the degree of (Q(x)) is written as

[dfrac{P(x)}{Q(x)}=dfrac{A_1x+B_1}{(a_1x^2+b1_x+c_1)}+dfrac{A_2x+B_2}{(a_2x^2+b_2x+c_2)}+⋅⋅⋅+dfrac{A_nx+B_n}{(a_nx^2+b_nx+c_n)}]

The decomposition may contain more rational expressions if there are linear factors. Each linear factor will have a different constant numerator: (A), (B), (C), and so on.

Howto: decompose a rational expression where the factors of the denominator are distinct, irreducible quadratic factors

- Use variables such as (A), (B), or (C) for the constant numerators over linear factors, and linear expressions such as (A_1x+B_1), (A_2x+B_2), etc., for the numerators of each quadratic factor in the denominator.
(dfrac{P(x)}{Q(x)}=dfrac{A}{ax+b}+dfrac{A_1x+B_1}{(a_1x^2+b1_x+c_1)}+dfrac{A_2x+B_2}{(a_2x^2+b_2x+c_2)}+⋅⋅⋅+dfrac{A_nx+B_n}{(a_nx^2+b_nx+c_n)})

- Multiply both sides of the equation by the common denominator to eliminate fractions.
- Expand the right side of the equation and collect like terms.
- Set coefficients of like terms from the left side of the equation equal to those on the right side to create a system of equations to solve for the numerators.

Example (PageIndex{3}): Decomposing (frac{P(x)}{Q(x)}) When (Q(x)) Contains a Nonrepeated Irreducible Quadratic Factor

Find a partial fraction decomposition of the given expression.

(dfrac{8x^2+12x−20}{(x+3)(x^2+x+2)})

**Solution**

We have one linear factor and one irreducible quadratic factor in the denominator, so one numerator will be a constant and the other numerator will be a linear expression. Thus,

(dfrac{8x^2+12x−20}{(x+3)(x^2+x+2)}=dfrac{A}{(x+3)}+dfrac{Bx+C}{(x^2+x+2)})

We follow the same steps as in previous problems. First, clear the fractions by multiplying both sides of the equation by the common denominator.

[egin{align*} (x+3)(x^2+x+2)left[dfrac{8x^2+12x-20}{(x+3)(x^2+x+2)} ight]&= left[dfrac{A}{(x+3)}+dfrac{Bx+C}{(x^2+x+2)} ight](x+3)(x^2+x+2)[4pt] 8x^2+12x-20&= A(x^2+x+2)+(Bx+C)(x+3) end{align*}]

Notice we could easily solve for (A) by choosing a value for (x) that will make the (Bx+C) term equal (0). Let (x=−3) and substitute it into the equation.

[egin{align*} 8x^2+12x-20&= A(x^2+x+2)+(Bx+C)(x+3)[4pt] 8{(-3)}^2+12(-3)-20&= A({(-3)}^2+(-3)+2)+(B(-3)+C)((-3)+3)[4pt] 16&= 8A[4pt] A&= 2 end{align*}]

Now that we know the value of (A), substitute it back into the equation. Then expand the right side and collect like terms.

[egin{align*} 8x^2+12x-20&= 2(x^2+x+2)+(Bx+C)(x+3)[4pt] 8x^2+12x-20&= 2x^2+2x+4+Bx^2+3B+Cx+3C[4pt] 8x^2+12x-20&= (2+B)x^2+(2+3B+C)x+(4+3C) end{align*}]

Setting the coefficients of terms on the right side equal to the coefficients of terms on the left side gives the system of equations.

Solve for (B) using Equation ef{3.1}

[egin{align*} 2+B&= 8 label{1} [4pt] B&= 6end{align*}

and solve for (C) using Equation ef{3.3}.

Thus, the partial fraction decomposition of the expression is

[dfrac{8x^2+12x−20}{(x+3)(x^2+x+2)}=dfrac{2}{(x+3)}+dfrac{6x−8}{(x^2+x+2)} onumber]

Q&A: Could we have just set up a system of equations to solve the example above?

Yes, we could have solved it by setting up a system of equations without solving for (A) first. The expansion on the right would be:

[egin{align*} 8x^2+12x-20&= Ax^2+Ax+2A+Bx^2+3B+Cx+3C[4pt] 8x^2+12x-20&= (A+B)x^2+(A+3B+C)x+(2A+3C) end{align*}]

So the system of equations would be:

Exercise (PageIndex{3})

Find the partial fraction decomposition of the expression with a nonrepeating irreducible quadratic factor.

[dfrac{5x^2−6x+7}{(x−1)(x^2+1)} onumber]

- Answer
(dfrac{3}{x−1}+dfrac{2x−4}{x^2+1})

## Decomposing (frac{P(x)}{Q(x)}) When (Q(x)) Has a Repeated Irreducible Quadratic Factor

Now that we can decompose a simplified rational expression with an irreducible quadratic factor, we will learn how to do partial fraction decomposition when the simplified rational expression has repeated irreducible quadratic factors. The decomposition will consist of partial fractions with linear numerators over each irreducible quadratic factor represented in increasing powers.

DECOMPOSITION OF (frac{P(x)}{Q(x)}) WHEN (Q(X)) HAS A REPEATED IRREDUCIBLE QUADRATIC FACTOR

The partial fraction decomposition of (dfrac{P(x)}{Q(x)}), when (Q(x)) has a repeated irreducible quadratic factor and the degree of (P(x)) is less than the degree of (Q(x)), is

[dfrac{P(x)}{{(ax^2+bx+c)}^n}=dfrac{A_1x+B_1}{(ax^2+bx+c)}+dfrac{A_2x+B_2}{{(ax^2+bx+c)}^2}+dfrac{A_3x+B_3}{{(ax^2+bx+c)}^3}+⋅⋅⋅+dfrac{A_nx+B_n}{{(ax^2+bx+c)}^n}]

Write the denominators in increasing powers.

How to: decompose a rational expression that has a repeated irreducible factor

- Use variables like (A), (B), or (C) for the constant numerators over linear factors, and linear expressions such as (A_1x+B_1), (A_2x+B_2), etc., for the numerators of each quadratic factor in the denominator written in increasing powers, such as
(dfrac{P(x)}{Q(x)}=dfrac{A}{ax+b}+dfrac{A_1x+B_1}{(ax^2+bx+c)}+dfrac{A_2x+B_2}{{(ax^2+bx+c)}^2}+⋅⋅⋅+dfrac{A_nx+B_n}{{(ax^2+bx+c)}^n})

- Multiply both sides of the equation by the common denominator to eliminate fractions.
- Expand the right side of the equation and collect like terms.

Example (PageIndex{4}): Decomposing a Rational Function with a Repeated Irreducible Quadratic Factor in the Denominator

Decompose the given expression that has a repeated irreducible factor in the denominator.

(dfrac{x^4+x^3+x^2−x+1}{x{(x^2+1)}^2})

**Solution**

The factors of the denominator are (x), ((x^2+1)), and ({(x^2+1)}^2). Recall that, when a factor in the denominator is a quadratic that includes at least two terms, the numerator must be of the linear form (Ax+B). So, let’s begin the decomposition.

(dfrac{x^4+x^3+x^2−x+1}{x{(x^2+1)}^2}=dfrac{A}{x}+dfrac{Bx+C}{(x^2+1)}+dfrac{Dx+E}{{(x^2+1)}^2})

We eliminate the denominators by multiplying each term by (x{(x^2+1)}^2). Thus,

[egin{align*} x^4+x^3+x^2-x+1&= A{(x^2+1)}^2+(Bx+C)(x)(x^2+1)+(Dx+E)(x)[4pt] x^4+x^3+x^2-x+1&= A(x^4+2x^2+1)+Bx^4+Bx^2+Cx^3+Cx+Dx^2+Exqquad ext{Expand the right side.}[4pt] &= Ax^4+2Ax^2+A+Bx^4+Bx^2+Cx^3+Cx+Dx^2+Ex end{align*}]

Now we will collect like terms.

(x^4+x^3+x^2−x+1=(A+B)x^4+(C)x^3+(2A+B+D)x^2+(C+E)x+A)

Set up the system of equations matching corresponding coefficients on each side of the equal sign.

[egin{align*} A+B&= 1[4pt] C&= 1[4pt] 2A+B+D&= 1[4pt] C+E&= -1[4pt] A&= 1 end{align*}]

We can use substitution from this point. Substitute (A=1) into the first equation.

[egin{align*} 1+B&= 1[4pt] B&= 0 end{align*}]

Substitute (A=1) and (B=0) into the third equation.

Substitute (C=1) into the fourth equation.

Now we have solved for all of the unknowns on the right side of the equal sign. We have (A=1), (B=0), (C=1), (D=−1), and (E=−2). We can write the decomposition as follows:

(dfrac{x^4+x^3+x^2−x+1}{x{(x^2+1)}^2}=dfrac{1}{x}+dfrac{1}{(x^2+1)}−dfrac{x+2}{{(x^2+1)}^2})

Exercise (PageIndex{4})

Find the partial fraction decomposition of the expression with a repeated irreducible quadratic factor.

[dfrac{x^3−4x^2+9x−5}{{(x^2−2x+3)}^2} onumber]

- Answer
[dfrac{x−2}{x^2−2x+3}+dfrac{2x+1}{{(x^2−2x+3)}^2} onumber]

## Key Concepts

- Decompose (dfrac{P(x)}{Q(x)}) by writing the partial fractions as [dfrac{A}{a_1x+b_1}+dfrac{B}{a_2x+b_2}. onumber] Solve by clearing the fractions, expanding the right side, collecting like terms, and setting corresponding coefficients equal to each other, then setting up and solving a system of equations (see Example (PageIndex{1})).
- The decomposition of (dfrac{P(x)}{Q(x)}) with repeated linear factors must account for the factors of the denominator in increasing powers (see Example (PageIndex{2})).
- The decomposition of (dfrac{P(x)}{Q(x)}) with a nonrepeated irreducible quadratic factor needs a linear numerator over the quadratic factor, as in (dfrac{A}{x}+dfrac{Bx+C}{(ax^2+bx+c)}) (see Example (PageIndex{3})).
- In the decomposition of (dfrac{P(x)}{Q(x)}), where (Q(x)) has a repeated irreducible quadratic factor, when the irreducible quadratic factors are repeated, powers of the denominator factors must be represented in increasing powers as [dfrac{A_1x+B_1}{ax^2+bx+c}+dfrac{A_2x+B_2}{(ax^2+bx+c)^2}+⋅⋅⋅+dfrac{A_nx+B_n}{(ax^2+bx+c)^n} onumber] See Example (PageIndex{4}).

## Partial Fractions

**Partial fraction decomposition** is a technique used to write a rational function as the sum of simpler rational expressions.

Partial fraction decomposition is a useful technique for some integration problems involving rational expressions. Partial fraction decomposition is also useful for evaluating telescoping sums. It is the basis for a proof of Euler's formula by finding the antiderivative of a rational expression in two different ways.

#### Contents

## 11.4: Partial Fractions - Mathematics

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## 11.4: Partial Fractions - Mathematics

so that we can now say that a partial fractions decomposition for is

This concept can also be used with functions of . For example,

so that we can now say that a partial fractions decomposition for is

Of course, what we would like to be able to do is find a partial fractions decomposition for a given function. For example, what would be a partial fractions decomposition for ? Begin by factoring the denominator, getting

Now ASSUME that there are constants and so that

- 1. both and are polynomials (constants together with positive integer powers of only)

- 2. the degree (highest power of ) of is smaller than the degree of .

(Get a common denominator and add the fractions.)

Since the fractions in the above equation have the same denominators, it follows that their numerators must be equal. Thus,

The right-hand side of this equation can be considered a function of which is equal to 6 for all values of . In particular, it must also be true for specific values of . For example, if we choose to

We can now say that a partial fractions decomposition for is

It should be noted that and were chosen for use in equation (**) for their convenience of ``zeroing out" terms in the equation. However, any other two choices for will lead to the exact same values for and (after solving two equations with two unknowns). Try it. After getting familiar with this process, in order to save some time, get in the habit of going from the step at equation (*) directly to the step at equation (**). Here is another important point to consider when applying the method of partial fractions to the rational function . If the degree (highest power) of is equal to or greater than the degree of , then you must use polynomial division in order to rewrite the given rational function as the sum of a polynomial and a new rational function satisfying condition 2 above. For example, polynomial division leads to

where the rational function on the right-hand side of the equation satisfies condition 2. There are other points to consider. Recall that the complex number so that and . In addition, if two complex numbers are equal, then their real and complex components are equal. That is, if

Now let's do another example. Find a partial fractions decomposition for . Begin by factoring the denominator, getting

Now ASSUME that there are constants and so that

Since is an irreducible quadratic expression, assuming only that

is NOT GENERAL ENOUGH and will not always lead to a correct partial fractions decomposition. Continuing, we have

(Get a common denominator and add the fractions.)

Since the fractions in the above equation have the same denominators, it follows that their numerators must be equal. Thus,

This equation can be considered two functions of which are equal to each other for all values of . In particular, it must also be true for specific values of . For example, if we ``conveniently" choose to

We can now say that a partial fractions decomposition for is

If you choose to NOT use complex numbers to solve for the unknown constants in the previous example, using TWO other real values of instead of will lead to the exact same values for and . There is one final case to consider. How should repeated factors in the denominator be handled ? The following example illustrates the partial fractions decomposition of a rational function, where the linear factor is repeated three times and the irreducible quadratic factor is repeated twice. Thus,

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## Partial Fractions

Algebra is a major component of mathematics that is used to unify mathematics concepts. Algebra is built on experiences with numbers and operations along with geometry and data analysis. The word “algebra” is derived from the Arabic word “al-Jabr”. The Arabic mathematician Al-Khwarizmi has traditionally been known as the “Father of algebra”. Algebra is used to find perimeter, area, volume of any plane figure and solid.

An improper fraction can be expressed as the sum of an integral function and a proper fraction.

Process of writing a single fraction as a sum or difference of two or more simpler fractions is called splitting up into *partial fractions* .

Generally if *p* (*x*) and *q* (*x*) are two rational integral algebraic functions of *x* and the fraction p(x)/q(x) be expressed as the algebraic sum (or difference) of simpler fractions according to certain specified rules, then the fraction p(x)/q(x) is said to be resolved into partial fractions.

## Partial Fractions

An expression of the form (frac < f(x) >< g(x) >), where f(x) and g(x) are polynomial in x, is called a rational fraction.

**Proper rational functions:**Functions of the form (frac < f(x) >< g(x) >), where f(x) and g(x) are polynomials and g(x) ≠ 0, are called rational functions of x.

If degree of f(x) is less than degree of g(x),then is called a proper rational function.**Improper rational functions:**If degree of f(x) is greater than or equal to degree of g(x), then (frac < f(x) >< g(x) >) is called an improper rational function.**Partial fractions:**Any proper rational function can be broken up into a group of different rational fractions, each having a simple factor of the denominator of the original rational function. Each such fraction is called a partial fraction.

If by some process, we can break a given rational function (frac < f(x) >< g(x) >) into different fractions, whose denominators are the factors of g(x),then the process of obtaining them is called the resolution or decomposition of (frac < f(x) >< g(x) >) into its partial fractions.

### Different cases of partial fractions

**(1) When the denominator consists of non-repeated linear factors:**

To each linear factor (x – a) occurring once in the denominator of a proper fraction, there corresponds a single partial fraction of the form (frac < A >< x-a >), where A is a constant to be determined.

If g(x) = (x – a_{1})(x – a_{2})(x – a_{3}) ……. (x – a_{n}), then we assume that,

where A_{1}, A_{2}, A_{3}, ………. A_{n} are constants, can be determined by equating the numerator of L.H.S. to the numerator of R.H.S. (after L.C.M.) and substituting x = a_{1}, a_{2},…… a_{n}. **(2) When the denominator consists of linear factors, some repeated:**

To each linear factor (x – a) occurring r times in the denominator of a proper rational function, there corresponds a sum of r partial fractions.

Let g(x) = (x – a) k (x – a_{1})(x – a_{2}) ……. (x – a_{r}). Then we assume that

Where A_{1}, A_{2}, A_{3}, ………. A_{k} are constants. To determine the value of constants adopt the procedure as above. **(3) When the denominator consists of non-repeated quadratic factors:**

To each irreducible non repeated quadratic factor ax 2 + bx + c, there corresponds a partial fraction of the form (frac < Ax+B >< < a< x >^< 2 >+bx+c > >), where A and B are constants to be determined.

Example :

**(4) When the denominator consists of repeated quadratic factors:**

To each irreducible quadratic factor ax 2 + bx + c occurring r times in the denominator of a proper rational fraction there corresponds a sum of r partial fractions of the form.

where, A’s and B’s are constants to be determined.

### Partial fractions of improper rational functions

If degree of is greater than or equal to degree of g(x), then (frac < f(x) >< g(x) >) is called an **improper rational function** and every rational function can be transformed to a proper rational function by dividing the numerator by the denominator.

We divide the numerator by denominator until a remainder is obtained which is of lower degree than the denominator.

## Quadratic Factors

**Theorem**. Suppose that $f(x) = P(x)/Q(x)$, where $P(x)$ and $Q$ are polynomials with no common factors and with the degree of $P$ less than the degree of $Q$. If $Q$ is the product of irreducible quadratic factors, then for each factor of the form $(ax^2+bx+c)^n$, the partial fraction decomposition is the following sum of $n$ partial fractions: egin

**Example**. Evaluate $displaystyle int frac<8(x^2+4)>

**Solution**. We use the method of partial fractions and write egin

**Example**. Evaluate $displaystyle int frac<20x> <(x-1)(x^2+4x+5)>, dx$.

**Example**. Evaluate $displaystyle int frac<2>

**Example**. Evaluate $displaystyle int frac<3x^4+4x^3+16x^2+20x+9> <(x+2)(x^2+3)^2>, dx$.

Welcome to **advancedhighermaths.co.uk**

A sound understanding of **Partial Fractions** is essential to ensure exam success.

Study at Advanced Higher Maths level will provide excellent preparation for your studies when at university. Some universities may require you to gain a pass at AH Maths to be accepted onto the course of your choice. The AH Maths course is fast paced so please do your very best to keep on top of your studies.

For students looking for extra help with the AH Maths course you may wish to consider subscribing to the fantastic additional exam focused resources available in the Online Study Pack.

To access a wealth of additional **free resources by topic** please either use the above Search Bar or click HERE selecting on the topic you wish to study.

We hope you find this website useful and wish you the very best of success with your AH Maths course in 2021/22. Please find below:

### Advanced Higher Maths Resources

**1. About Partial Fractions **

To learn about Partial Fractions please click on any of the Theory Guide links in Section 2 below. For students working from the Maths In Action text book the recommended questions on this topic are given in Section 3. Worksheets including actual SQA Exam Questions are highly recommended.

If you would like more help understanding **Partial Fractions** there are full, easy to follow, step-by-step worked solutions to dozens of AH Maths Past & Practice exam questions on all topics in the AH Maths Online Study Pack. Also included in the Study Pack are full worked solutions to the recommended MIA text book questions. Please give yourself every opportunity for success, speak with your parents, and subscribe to the **exam focused** Online Study Pack today.

**Partial Fractions**

- Partial Fractions are a way of ‘breaking apart’ fractions with polynomials in them
- Some types of rational functions p(x)/q(x) can be decomposed into Partial Fractions
- If the numerator is of a higher (or equal) degree than the denominator, then algebraic long division should be used first to obtain a proper rational function
- In exams q(x) can be either a quadratic or cubic which can be factorised easily into one of three types – Linear Factors, Repeated Linear Factor or Irreducible Factor.

An example of each of the three types is shown below.

Example One – Distinct Linear Factors

Example Two – Repeated Linear Factor

If the denominator contains a repeated linear factor, more than one partial fraction must be included for this factor, as illustrated in the example below.

Example Three – Irreducible Factor

**Exam Question**

* *

Source: SQA AH Maths Paper 2017 Question 2

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**2. Partial Fractions – Exam Worksheet & Theory Guides**

Thanks to the SQA and authors for making the excellent AH Maths Worksheet & Theory Guides freely available for all to use. These will prove a fantastic resource in helping consolidate your understanding of AH Maths. Clear, easy to follow, step-by-step worked solutions to all SQA AH Maths Questions in the worksheet below are available in the Online Study Pack.

Worksheet/Theory Guides __________________________ | Resource Link ________________________________ | Answers ____________ |

AH Exam Questions | Partial Fractions Exam Questions | Answers |

AH Maths Formulae List | AH Maths Fomulae List | |

Theory Guide 1 | Partial Fractions Theory Guide 1 | |

Theory Guide 2 | Partial Fractions Theory Guide 2 | |

Theory Guide 3 (HSN) | Partial Fractions Theory Guide 3 (HSN) |

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**3. Partial Fractions – Recommended Text Book Questions**

Recommended questions from the Maths In Action (2nd Edition) by Edward Mullan text book are shown below. Clear, easy to follow, step-by-step worked solutions to all questions below are available in the Online Study Pack.

Subtopic _______________________________ | Page Number _____________ | Exercise _____________ | Recommended Questions _______________________ | Comment ________________ |

Type One - Partial Fractions | Page 23 | Exercise 2.2 | Q1, 5, 12, 18, 19, 22, 25 | |

Type Two - Partial Fractions | Page 24 | Exercise 2.3 | Q1, 3, 5, 10, 14, 18 | |

Type Three - Partial Fractions | Page 25 | Exercise 2.4 | Q1, 5, 7, 9, 11 | |

Algebraic Long Division Worksheet | Worksheet | Worked Solutions | ||

Partial Fraction - Long Division | Page 26 | Exercise 2.5 | Q1 a, b, e, j, l |

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**4. AH Maths Past Paper Exam Worksheets by Topic**

Thanks to the SQA for making these available. The worksheets by topic below are an excellent study resource since they are actual SQA past paper exam questions. Clear, easy to follow, step-by-step worked solutions to all SQA AH Maths Questions below are available in the Online Study Pack.

Number ______ | Topic ____________________________________________ | Answers _________ |

1 | Binomial Theorem | Answers |

2 | Complex Numbers | Answers |

3 | Differentiation | Answers |

4 | Differentiation (Further) | Answers |

5 | Differential Equations - Variables Separable | Answers |

6 | Differential Equations (Further) | Answers |

7 | Functions & Graphs | Answers |

8 | Integration | Answers |

9 | Integration (Further) | Answers |

10 | Matrices | Answers |

11 | Number Theory - Methods of Proof | Answers |

12 | Number Theory (Further) - Euclidean & Number Bases | Answers |

13 | Partial Fractions | Answers |

14 | Sequences & Series | Answers |

15 | Sequences & Series - Maclaurin | Answers |

16 | Systems of Equations | Answers |

17 | Vectors | Answers |

**5. AH Maths Past Paper Questions by Topic**

Thanks to the SQA for making these available. Questions and answers have been split up by topic for your ease of reference. Clear, easy to follow, step-by-step worked solutions to all SQA AH Maths questions below are available in the Online Study Pack.

. Paper ___________ | . Marking ______ | Binomial Theorem ________ | Partial Fractions ________ | . Differentiation ___________ | Further Differentiation ___________ | . Integration ___________ | Further Integration ____________ | Functions & Graphs ___________ | Systems of Equations ____________ | Complex Numbers __________ | Seq & Series _________ | Further Seq & Series ____________ | . Matrices _________ | . Vectors __________ | Methods of Proof __________ | Further No Theory ___________ | Differential Equations ____________ | Further Differential Eqns _________________ |

Specimen P1 | Marking | Q2 | Q4 | Q6 | Q8 | Q3 | Q5 | Q1 | Q7 | |||||||||

Specimen P2 | Marking | Q3 | Q1 | Q2,4,8,10 | Q7 | Q11 | Q5 | Q13 | Q9 | Q6 | Q12 | |||||||

2019 | Marking | Q9 | Q4 | Q1a,b,6 | Q1c,5,10 | Q16b | Q16a | Q3 | Q18 | Q7,17 | Q2 | Q15 | Q11,14 | Q12 | Q13 | Q8 | ||

2018 | Marking | Q3 | Q2 | Q1b | Q1a,c,6,13 | Q8 | Q15a | Q16a | Q4,10 | Q14 | Q17 | Q7,11 | Q16 | Q9,12 | Q5 | Q15b | ||

2017 | Marking | Q1 | Q2 | Q3 | Q11,18 | Q16 | Q6 | Q12 | Q5 | Q17 | Q4,10 | Q7 | Q15 | Q13 | Q8 | Q9 | Q14 | |

2016 | Marking | Q3 | Q13 | Q1a,b | Q1c,11 | Q13 | Q9 | Q12 | Q4 | Q8 | Q2 | Q6 | Q7 | Q14 | Q5,10 | Q16 | Q15 | |

2015 | Marking | Q1,9 | Q2 | Q4,6,8 | Q17 | Q10 | Q14 | Q13 | Q3 | Q5,11 | Q15 | Q12 | Q7 | Q18 | Q16 | |||

2014 | Marking | Q2 | 14b | Q1,13 | Q1,4,6 | Q10,12 | Q15 | Q11 | Q3 | Q16 | Q14 | Q9 | Q7 | Q5 | Q7 | Q8 | ||

2013 | Marking | Q1 | Q2 | Q11 | Q4,6 | Q8 | Q13 | Q7,10 | Q17 | Q3 | Q15 | Q9,12 | Q5 | Q16 | Q14 | |||

2012 | Marking | Q4 | 15a | Q1 | Q12,13 | Q8 | Q11 | Q7 | Q14 | Q3,16b | Q2 | Q6 | Q9 | Q5 | 16a | Q10 | Q15 | |

2011 | Marking | Q2 | Q1 | 3b,7 | 3a | Q1,11a | Q1,11,16 | Q6 | Q10 | Q8,13 | Q5 | Q4 | Q15 | Q12 | Q9 | Q14 | ||

2010 | Marking | Q5 | Q1 | Q13 | Q15 | Q3,7 | Q10 | Q16 | Q2 | Q9 | Q4,14 | Q6 | Q8,12 | Q11 | ||||

2009 | Marking | Q8 | Q14 | Q1a | Q1b,11 | Q5,7 | Q9 | Q13 | 16a | Q6 | Q12 | Q14 | Q2 | Q16 | Q4 | Q10 | Q3 | Q15 |

2008 | Marking | Q8 | Q4 | Q10,15 | Q2,5 | Q4,9,10 | Q7 | Q3 | Q16 | Q1 | Q12 | Q6 | Q14 | Q11 | Q13 | |||

2007 | Marking | Q1 | Q4 | Q2 | Q13 | Q4,10 | Q4 | Q16 | Q3,11 | Q9 | Q6 | Q5 | Q15 | Q12 | Q7 | Q14 | Q8 | |

Mixed | Mixed | Mixed | Mixed | Mixed | Mixed | Mixed | Mixed | Mixed | Mixed | Mixed | Mixed | Mixed | Mixed | Mixed | Mixed | Mixed |

**6. AH Maths Past & Practice Exam Papers**

Thanks to the SQA for making these available. Clear, easy to follow, step-by-step worked solutions to all SQA AH Maths questions below are available in the Online Study Pack.

Year ____ | Paper Type _________________ | Exam Paper ______________ | Marking Scheme _______________________________________ |

2019 | AH Specimen | Specimen | Marking Scheme |

2019 | Advanced Higher | Exam Paper | Marking Scheme |

2018 | Advanced Higher | Exam Paper | Marking Scheme |

2017 | Advanced Higher | Exam Paper | Marking Scheme |

2016 | Advanced Higher | Exam Paper | Marking Scheme |

2016 | AH Specimen | Specimen | Marking Scheme |

2016 | AH Exemplar | Exemplar | Marking Scheme |

2015 | Advanced Higher | Exam Paper | Marking Scheme |

2014 | Advanced Higher | Exam Paper | Marking Scheme |

2013 | Advanced Higher | Exam Paper | Marking Scheme |

2012 | Advanced Higher | Exam Paper | Marking Scheme |

2011 | Advanced Higher | Exam Paper | Marking Scheme |

2010 | Advanced Higher | Exam Paper | Marking Scheme |

2009 | Advanced Higher | Exam Paper | Marking Scheme |

2008 | Advanced Higher | Exam Paper | Marking Scheme |

2007 | Advanced Higher | Exam Paper | Marking Scheme |

2006 | Advanced Higher | Exam Paper | Marking Scheme |

2005 | Advanced Higher | Exam Paper | Marking Scheme |

2004 | Advanced Higher | Exam Paper | Marking Scheme |

2003 | Advanced Higher | Exam Paper | Marking Scheme |

2002 | Advanced Higher | Exam Paper | Marking Scheme |

2001 | Advanced Higher | Exam Paper | Marking Scheme |

**7. AH Maths 2020 Specimen Exam Paper**

Please find below two Specimen Papers courtesy of the SQA. Clear, easy to follow, step-by-step worked solutions to the SQA AH Maths Specimen Paper available in the Online Study Pack.

. Date __________ | . Paper ___________ | . Marking ______ | Binomial Theorem ________ | Partial Fractions ________ | . Differentiation ___________ | Further Differentiation ___________ | . Integration ___________ | Further Integration ____________ | Functions & Graphs ___________ | Systems of Equations ____________ | Complex Numbers __________ | Seq & Series _________ | Further Seq & Series ____________ | . Matrices _________ | . Vectors __________ | Methods of Proof __________ | Further No Theory ___________ | Differential Equations ____________ | Further Differential Eqns _________________ |

June 2019 | Specimen P1 | Marking | Q2 | Q4 | Q6 | Q8 | Q3 | Q5 | Q1 | Q7 | |||||||||

June 2019 | Specimen P2 | Marking | Q3 | Q1 | Q2,4,8,10 | Q7 | Q11 | Q5 | Q13 | Q9 | Q6 | Q12 |

.

**8. AH Maths Prelim & Final Exam Practice Papers**

Thanks to the SQA and authors for making these freely available. Please use regularly for revision prior to assessments, tests and the final exam. Clear, easy to follow, step-by-step worked solutions to the first five Practice Papers below are available in the Online Study Pack.

AH Practice Exam Paper _____________________ | Marking ___________ | AH Practice Exam Paper _____________________ | Marking ___________ |

Practice Exam Paper 1 | HERE | Practice Exam Paper 5 | HERE |

Practice Exam Paper 2 | HERE | Practice Exam Paper 6 | HERE |

Practice Exam Paper 3 | HERE | Practice Exam Paper 7 | HERE |

Practice Exam Paper 4 | HERE | Practice Exam Paper 8 | HERE |

**9. AH Maths Theory Guides**

Thanks to the authors for making the excellent AH Maths Theory Guides freely available for all to use. These will prove a fantastic resource in helping consolidate your understanding of AH Maths.

Topic 1 ______________________ | Topic 2 ___________________ | Topic 3 _____________________ | Topic 4 ___________________ | Topic 5 ___________________ | Topic 6 ___________________ |

Partial Fractions 1 | Binomial 1 | Gaussian 1 | Functions 1 | Differentiation 1 | Integration 1 |

Partial Fractions 2 | Binomial 2 | Gaussian 2 | Functions (HSN) | Differentiation 2 | Integration (HSN) |

Partial Fractions (HSN) | Binomial (HSN) | Gaussian (HSN) | Differentiation (HSN) |

Topic 1 ______________________ | Topic 2 ________________________ | Topic 3 ___________________ | Topic 4 ____________________ | Topic 5 _________________________ |

Further Differentiation 1 | Further Integration 1 | Complex Numbers 1 | Sequences & Series 1 | Methods of Proof |

Further Differentiation 2 | Further Integration 2 | Complex Numbers 2 | Sequences & Series 2 | Proof by Induction |

Differentiation (HSN) | Integration (HSN) | Complex Nos (HSN) | Seq & Series (HSN) | Methods of Proof (HSN) |

Topic 1 ________________________ | Topic 2 _________________ | Topic 3 _____________________ | Topic 4 _____________________ | Topic 5 ______________________________ |

Vectors 1 | Matrices 1 | Maclaurin Series 1 | Differential Eqns 1 | Further Number Theory |

Vectors 2 | Matrices 2 | MacLaurin Series 2 | Differential Eqns 2 | |

Vectors 3 | Matrices 3 | Maclaurin Series (HSN) | Differential Eqns (HSN) | |

Vectors (HSN) | Matrices (HSN) |

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**10. AH Maths Course Outline, Formulae Sheets & Check List**

Thanks to the SQA and authors for making the excellent resources below freely available. These are fantastic check lists to assess your AH Maths knowledge. Please try to use these regularly for revision prior to tests, prelims and the final exam.

Title ____________________________________ | Link ___________ | Courtesy ___________________ |

AH Maths Course Outline & Timings | HERE | |

SQA AH Maths Exam Formulae List | HERE | Courtesy of SQA |

SQA Higher Maths Exam Formulae List | HERE | Courtesy of SQA |

SQA AH Maths Support Notes | HERE | Courtesy of SQA |

AH Maths Complete Check List | HERE |

**11. Text Book Recommended Timings & Questions – Unit One**

Course timings, along with specific text book exercises/questions for Unit One, courtesy of Teejay Publishers can be found HERE .

**Partial Fractions**

Recommended questions from the Maths In Action (2nd Edition) by Edward Mullan Text Book are shown below. Clear, easy to follow, step-by-step worked solutions to all questions below are available in the Online Study Pack.

Subtopic _______________________________ | Page Number _____________ | Exercise _____________ | Recommended Questions _______________________ | Comment ________________ |

Type One - Partial Fractions | Page 23 | Exercise 2.2 | Q1, 5, 12, 18, 19, 22, 25 | |

Type Two - Partial Fractions | Page 24 | Exercise 2.3 | Q1, 3, 5, 10, 14, 18 | |

Type Three - Partial Fractions | Page 25 | Exercise 2.4 | Q1, 5, 7, 9, 11 | |

Algebraic Long Division Worksheet | Worksheet | Worked Solutions | ||

Partial Fraction - Long Division | Page 26 | Exercise 2.5 | Q1 a, b, e, j, l |

**Binomial Theorem**

Recommended questions from the Maths In Action (2nd Edition) by Edward Mullan Text Book are shown below. Clear, easy to follow, step-by-step worked solutions to all questions below are available in the Online Study Pack.

Subtopic ____________________________________ | Page Number _____________ | Exercise ___________ | Recommended Questions _______________________________ | Notes for Lesson __________________________________________________________________________________ |

Combinations nCr | Page 33 | Exercise 3.3 | Q1a,b,c,2a,b,c,4a-d,5a,b,6a,7a,b,d | |

Expanding - Lesson 1 | Page 36 | Exercise 3.4 | Q1a,b,c,2a,i,ii,iii,iv | |

Expanding - Lesson 2 | Page 36 | Exercise 3.4 | Q3a-d,4a-f | THEORY - Questions 3 & 4 |

Finding Coefficients | Page 38 | Exercise 3.5 | Q1a,b,c,4a,5a,6 | |

Approximation eg 1.05^5 = ? | Page 40 | Exercise 3.6 | Q1a,b,c,d | |

Simplifying General Term (SQA Questions) | SQA Questions & Answers | Common SQA Binomial Questions not in AH Text Book |

**Systems of Equations**

Recommended questions from the Maths In Action (2nd Edition) by Edward Mullan Text Book are shown below. Clear, easy to follow, step-by-step worked solutions to all questions below are available in the Online Study Pack.

Subtopic ______________________________ | Page Number _____________ | Exercise _______________ | Recommended Questions _______________________ |

Gaussian Elimination | Page 265 | Exercise 14.4 | Q1a,b,c,d,2a,b,c |

Redundancy & Inconsistency | Page 268 | Exercise 14.6 | Q1a,b,c,2 |

Redundancy SQA Question | 2016 Q4 (SQA) | ||

Inconsistency SQA Question | 2017 Q5 (SQA) | ||

ILL Conditioning | Page 274 | Exercise 14.9 | Q2a,b,c,d |

ILL Conditioning SQA Question | 2012 Q14c (SQA) |

**Functions & Graphs**

Subtopic ______________________________ | Page Number _____________ | Exercise ___________ | Recommended Questions _______________________ |

Sketching Modulus Function y = |x| | Page 66 | Exercise 5.2 | Q1-9 |

Inverse Functions | Page 67 | Exercise 5.3 | Q1a,c,e,g,i,2a,c,e,3 |

Odd & Even Functions | Page 74 | Exercise 5.8 | Q3a-l |

Vertical Asymptotes & Behaviour | Page 75 | Exercise 5.9 | Q1a-f |

Horizontal & Oblique Asymptotes | Page 76 | Exercise 5.10 | Q1a,b,f,g,k,l |

Sketching Graphs | Page 77 | Exercise 5.11 | Q1a,c,e,i,k |

**Differential Calculus**

Subtopic ___________________________ | Page Number ____________ | Exercise ___________ | Recommended Questions _______________________ |

Derivative from First Principles | Page 45 | Exercise 4.1 | Q1,3,5,7 |

The Chain Rule | Page 48 | Exercise 4.3 | Q1a,d,2a,c,3b,4a,5a |

The Product Rule | Page 51 | Exercise 4.5 | Q1a-h,Q2b,Q3a-l |

The Quotient Rule | Page 52 | Exercise 4.6 | Q1,2,3,4 |

Differentiation - A Mixture! | Page 53 | Exercise 4.7 | Q1,2,3,4,5 |

Sec, Cosec & Cot | Page 55 | Exercise 4.8 | Q1a,b,2a,c,d,3a,c,e,g |

Exponential Functions | Page 58 | Exercise 4.9 | Q1a,c,e,2a,3e,4a,b,5a,e |

Logarithmic Functions | Page 58 | Exercise 4.9 | Q1k,m,o,q,s,2f,g,3a,b,c,4d,e,5d |

Nature & Sketching Polynomials | Page 70 | Exercise 5.5 | Q1a,b,c,2a,b |

Concavity | Page 73 | Exercise 5.7 | Q5a,b,c,Q1a,b |

Applications | Page 187 | Ex 11.1 | Q1a,b,e,f,2a,c,3a,c |

**Integral Calculus**

Subtopic ______________________________________ | Page No __________ | Exercise ___________ | Recommended Questions _____________________ |

Integration (Higher Revision) | Page 100 | Exercise 7.1 | Q1a-i,2a-i,3a-l,4a-f |

Integration by Substitution | Page 103 | Exercise 7.2 | Q1a,c,e,g,i,k,m,o,q,s,u,w |

Integration by Substitution - Extra Revision! | Page 103 | Exercise 7.2 | Q1b,d,f,h,j,l,n,p,r,t,v,x |

Further Integration by Substitution | Page 105 | Exercise 7.3 | Q2a,b,c,d,4a,b,c,d |

Further Integration by Substitution | Page 105 | Exercise 7.3 | Q6a,b,c,d |

Further Int'n by Sub'n - sin^m(x), cos^n(x) | Page 105 | Exercise 7.3 | Q7a,b,c,d,e,f |

Further Integration by Substitution - logs | Page 105 | Exercise 7.3 | Q11a,b,c,d |

Substitution & Definite Integrals | Page 107 | Exercise 7.4 | Q1a,c,e,g,i,k |

Area between curve & x-axis | Page 120 | Exercise 7.10 | Q1,3 |

Area between curve & y-axis | Page 120 | Exercise 7.10 | Q6,7 |

Volume - revolved around x-axis SQA Question | 2014 Q10 (SQA) | ||

Volume - revolved around y-axis SQA Question | 2017 Q16 (SQA) | ||

Volume - revolved around x-axis | Page 120 | Exercise 7.10 | Q11,12 |

Applications of Integral Calculus | Page 187 | Exercise 11.1 | Q4,14 |

**12. Text Book Recommended Timings & Questions – Unit Two**

Course timings, along with specific text book exercises/questions for Unit Two, courtesy of Teejay Publishers can be found HERE .

**Further Differentiation**

Subtopic _______________________________________ | Page Number _____________ | Exercise ______________ | Recommended Questions _____________________ |

Inverse Trig Functions & Chain Rule | Page 85 | Exercise 6.2 | Q1a,b,c,Q2b,c,dQ3a,d |

Inverse Trig Fns & Product/Quotient Rules | Page 86 | Exercise 6.3 | Q2,Q3 |

Implicit & Explicit Functions - 1 | Page 89 | Exercise 6.4 | Q1,Q2 |

Implicit & Explicit Functions - 2 | Page 89 | Exercise 6.4 | Q5,Q9,Q4 |

Second Derivatives of Implicit Functions | Page 90 | Exercise 6.5 | Q1a,d,f,k(i),6 |

Logarithmic Differentiation | Page 92 | Exercise 6.6 | Q1,Q2 |

Parametric Equations | Page 95 | Exercise 6.7 | Q1a,b,c |

Parametric Eqns - Differentiation | Page 96 | Exercise 6.8 | Q1,2,3 |

Parametric Eqns - Differentiation (Alternative) | Page 96 | Exercise 6.8 | Q1(i) |

Parametric Eqns - Differentiation (Alternative) | Page 96 | Exercise 6.8 | Q1(ii),Q2,Q3 |

Applications of Further Differentiation | Page 193 | Exercise 11.2 | Q1,Q2,Q3 |

**Further Integration**

Recommended questions from the Maths In Action (2nd Edition) by Edward Mullan Text Book shown below. Clear, easy to follow, step-by-step worked solutions to all questions below are available in the Online Study Pack.

Subtopic ______________________________________ | Page No __________ | Exercise _________________ | Recommended Questions __________________________ |

Integration using Inverse Trig Functions | Page 111 | Exercise 7.6 | Q1,2,3,4a,b |

Integration using Partial Fractions | Page 113 | Exercise 7.7 | Q1a,b,2a,b,3a,b,4a,b,5a,b,6a,b |

Integration by Parts - 1 | Page 116 | Exercise 7.8 | Q1a-l |

Integration by Parts - 2 | Page 116 | Exercise 7.8 | Q2a,c,d,e,f,g,h |

Integration by Parts - 3 | Page 116 | Exercise 7.8 | Q5a,b,Q6a,b |

Integration by Parts - Special Cases - 1 | Page 118 | Exercise 7.9 | Q1a,b,c,d |

Integration by Parts - Special Cases - 2 | Page 118 | Exercise 7.9 | Q2a,b,c,d,e |

First Order Diff Eqns - General Soln | Page 128 | Exercise 8.1 | Q1a-j |

First Order Diff Eqns - Particular Soln | Page 128 | Exercise 8.1 | Q2a-g |

Differential Equations in Context | Page 131 | Exercise 8.2 | Q2,4,5,6 |

**Complex Numbers**

Subtopic _________________________________ | Page Number _____________ | Exercise ______________ | Recommended Questions _________________________ |

Arithmetic with Complex Numbers | Page 207 | Exercise 12.1 | Q1,2,3,6,7,8 |

Division & Square Roots of Complex Nos | Page 209 | Exercise 12.2 | Q1a,b,c,2c,e,3a,b,f,5a,b |

Argand Diagrams | Page 211 | Exercise 12.3 | Q3a,b,d,e,f,i,6a,b,f,7a,b,c |

Multiplying/Dividing in Polar Form | Page 215 | Exercise 12.5 | Q1a,b,f,g |

De Moivre's Theorem | Page 218 | Exercise 12.6 | Q1,2,3a,4g,h,i,j |

Polynomials & Complex Numbers | Page 224 | Exercise 12.8 | Q2a,d,3a,b,4,5,6a,b |

Loci on the Complex Plane | Page 213 | Exercise 12.4 | Q1a,b,d,f,j,3a,b,4a,b,c |

Expanding Trig Formula | Page 219 | Exercise 12.6 | Q5,6,7a |

Roots of a Complex Number | Page 222 | Exercise 12.7 | Q2a,b,c,d,e,f,1a(i) |

**Sequences & Series, Sigma Notation**

Subtopic _______________________________ | Page Number _____________ | Exercise ______________ | Recommended Questions __________________________ |

Arithmetic Sequences | Page 151 | Exercise 9.1 | Q1a-f,2a-f,Q3,Q4,Q6 |

Finding Sum - Arithmetic Sequence | Page 153 | Exercise 9.2 | Q1a,b,c,Q3a-d,Q4a,b,Q5a |

Geometric Sequence | Page 156 | Exercise 9.3 | Q1a-e,Q2,Q3,Q5 |

Finding Sum - Geometric Sequence | Page 159 | Exercise 9.4 | Q1a-f,Q2a-d,Q3a-d,Q4 |

Finding Sum to Infinity | Page 162 | Exercise 9.5 | Q1,2,3,4,6 |

Sigma Notation | Page 168 | Exercise 10.1 | Q1a-e,Q2a-e |

**Number Theory & Proof**

Topic _______________________________ | Lessons __________ | Questions _________ | Typed Solutions _______________ | Handwritten Solutions ______________________ | Exam Questions - Worked Solutions in Online Study Pack ______________________________________________________ |

Direct Proof | Lesson 1 | Ex 1 & 2 | Ex 1 & 2 Handwritten Solns | 2018-Q9,2015-Q12, 2010-Q8a | |

Proof by Counterexample | Lesson 2 | Ex 3 | Ex 3 Typed Solns | Ex 3 Handwritten Solns | 2016-Q10, 2013-Q12, 2008-Q11 |

Proof by Counterexample | Ex 4 | Ex 4 Typed Solns | Ex 4 Handwritten Solns | 2016-Q10, 2013-Q12, 2008-Q11 | |

Proof by Contradiction | Lesson 3 | Ex 5 | Ex 5 Typed Solns | Ex 5 Handwritten Solns | 2010-Q12 |

Proof by Contrapositive | Lesson 4 | Ex 6 | Ex 6 Typed Solns | Ex 6 Handwritten Solns | 2017-Q13 |

Proof by Induction | Lesson 5 | Ex 7 | Ex 7 Typed Solns | Ex 7 Handwritten Solns | 2014-Q7,2013-Q9,2012-Q16a,2011-Q12,2010-Q8b,2009-Q4,2007-Q12 |

Proof by Induction - Sigma Notation | Lesson 6 | Ex 8 | Ex 8 Typed Solns | Ex 8 Handwritten Solns | 2018-Q12,2016-Q5, 2013-Q9,2009-Q4 |

**13. Text Book Recommended Timings & Questions – Unit Three**

Course timings, along with specific text book exercises/questions for Unit Three, courtesy of Teejay Publishers can be found HERE .

Subtopic __________________________________ | Page Number _____________ | Exercise ______________ | Recommended Questions ________________________ | Lesson/Notes _________________ |

Higher Revision On Vectors | Page 282 | Exercise 15.1 | Q6,7,8 | |

The Vector Product - 1 | Page 286 | Exercise 15.3 | Q1,2a,b,5,7,8a,b,10 | Lesson 1 |

The Vector Product - 2 | Page 286 | Exercise 15.3 | Q3,4,6,12 | Lesson 2 |

The Equations of a Line | Page 298 | Exercise 15.8 | Q1a,b,2a,3a,c,e,5 | Lesson 3 |

Vector Equation of a Straight Line | Page 298 | Exercise 15.9 | Q2 | Lesson 3 |

The Equation of a Plane | Page 291 | Exercise 15.5 | Q1a,b,c,d,2a,b,3,4a,c,9,10 | Lesson 4 |

Angle Between 2 Planes | Page 293 | Exercise 15.6 | Q1,2,3 | Lesson 5 |

Intersection of Line & Plane | Page 300 | Exercise 15.10 | Q1a,b,c,2a,b,3,4a | Lesson 6 |

Intersection of 2 Lines | Page 302 | Exercise 15.11 | Q1,2 | Lesson 7 |

Intersection of 2 Planes using Gaussian | Page 303 | Exercise 15.12 | Q1,2 | Lesson 8 |

Intersection of 2 Planes - Alternative | Page 303 | Exercise 15.12 | Q1,2 | |

Intersection of 3 Planes | Page 307 | Exercise 15.3 | Q1a,c,2a,c | Lesson 9 |

Subtopic __________________________________ | Page Number _____________ | Exercise ______________ | Recommended Questions ____________________________ |

Basic Properties & Operations of Matrices | Page 231 | Exercise 13.1 | Q1,2,3a,4a,c,e,i,p,t,7a,f,9,10 |

Matrix Multiplication | Page 235 | Exercise 13.3 | Q1a,c,2a,c,k,m,o,3a,4,5a,c |

Properties of Matrix Multiplication | Page 236 | Exercise 13.4 | Q6a,b,7a,b,8a |

Determinant of a 2 x 2 Matrix | Page 240 | Exercise 13.6 | Q1a,b,d,h |

Determinant of a 3 x 3 Matrix | Page 247 | Exercise 13.9 | Q4a,b,c,d,5a,b |

Inverse of a 2 x 2 Matrix | Page 243 | Exercise 13.7 | Q1,2,4,8,9a,b,c |

Inverse of a 3 x 3 Matrix | Page 275 | Exercise 14.10 | Q1a,b,c,d |

Transformation Matrices | Page 251 | Exercise 13.10 | Q1,2,5 |

**Further Sequences & Series (Maclaurin Series)**

Subtopic ________________________________ | Page Number _____________ | Exercise ______________ | Recommended Questions _______________________ |

Maclaurin Series for f(x) | Page 179 | Exercise 10.5 | Q1a,b,c,d,3a,b |

Maclaurin Series - Composite Functions | Page 182 | Exercise 10.7 | Q1a,f,2a,3a,6a,7a,8a,b |

Maclaurin Series - SQA Questions | SQA Questions & Answers |

**Further Differential Equations**

Subtopic __________________________________ | Page Number _____________ | Exercise ______________ | Recommended Questions ________________________ |

1st Order Linear Differential Equations | Page 136 | Exercise 8.3 | Q1a,b,2a,3a,b |

2nd Order Differential Equations (Roots Real & Distinct) | Page 140 | Exercise 8.4 | Q1a,b,c,2a,b |

2nd Order Differential Equations (Roots Real & Coincident) | Page 141 | Exercise 8.5 | Q1a,b,c,2a,b |

2nd Order Differential Equations (Roots Not Real) | Page 142 | Exercise 8.6 | Q1a,b,c,2a,b |

Non-Homogeneous Differential Equations (Finding General Solution) | Page 146 | Exercise 8.9 | Q1a,b,c |

Non-Homogeneous Differential Equations (Finding Particular Solution) | Page 146 | Exercise 8.9 | Q2a,b,c |

**Further Number Theory & Proof**

Subtopic _______________________________________ | Page Number _____________ | Exercise _________ | Recommended Questions ____________________________ |

Finding the Greatest Common Divisor (GCD) | Page 318 | Ex 16.3 | Q1a,c,e,g,i |

Expressing GCD in the form xa + yb = d | Page 320 | Ex 16.4 | Q1,2,3,4 |

Number Bases | Page 322 | Ex 16.5 | Q1a-d,2a-f |

Further Number Theory - SQA Questions | SQA Questions & Answers |

**14. AH Maths Practice Unit Assessments – Solutions Included**

Thanks to maths777 for making the excellent resources freely available for all to use. This will prove a fantastic resource in helping you prepare for assessments, tests and the final exam.

Methods in Algebra & Calculus __________________________ | Applications of Algebra & Calculus ____________________________ | Geometry, Proof & Systems of Equations ____________________________________ |

Practice 1 | Practice 1 | Practice 1 |

Practice 2 | Practice 2 | Practice 2 |

Practice 3 | Practice 3 | Practice 3 |

**15. AH Maths Video Links**

Please click DLB Maths to view AH Maths Past Paper video solutions. There are also many videos showing worked examples by topic on the St Andrews StAnd Maths YouTube Channel link. Both video links are excellent resources in helping you prepare for assessments, tests and the final exam.

**16.** **AH Maths Text Book – Maths In Action (2nd Edition) by Edward Mullan**

A fully revised course for the new Curriculum for Excellence examination that is designed to fully support the course’s new structure and unit assessment. A part of the highly regarded Maths in Action series, it provides students with a familiar, clear and carefully structured learning experience that encourages them to build confidence and understanding.

**17. Advanced Higher Maths Online Study Pack** ** **

Through step-by-step worked solutions to exam questions and recommended MIA text book questions available in the Online Study Pack we cover everything you need to know about **Partial Fractions** to pass your final exam.

For students looking for a ‘good’ pass at AH Maths you may wish to consider subscribing to the fantastic additional exam focused resources available in the Online Study Pack. Subscribing could end up being one of your best ever investments.

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## Introduction

To compute the coefficients using the cover-up method, first set up a partial fraction decomposition with one term for each of the factors in the denominator. For example, if the denominator has three distinct linear terms, we have the decomposition

Note:Keep in mind that in order to apply partial fractions, the degree of the polynomial in the numerator must be strictly smaller than the degree of the polynomial in the denominator. If this is not the case, then it is necessary to first apply polynomial division to obtain a quotient polynomial and a remainder, for which the degree of the numerator is strictly smaller than that of the denominator. Partial fractions may then be applied to the remainder.

Here is a basic example on how to use the partial fraction rule for factorization.

Try the following problem based on the understanding of applying partial fraction.

The equation above represents a partial fraction decomposition for constants A , B , C A,B,C A , B , C and P P P .

What is the smallest value of the prime number P P P such that A , B A,B A , B and C C C are all integers?

## 11.4: Partial Fractions - Mathematics

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##### Updated Version Available

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##### Mathematical Expression Editor

In Section ?? we saw that expansions into partial fractions is a necessary tool when applying the method of Laplace transforms. In the simplest case partial fractions work as follows. Assume that and are two polynomials such that

(a) the degree of is less or equal to the degree of (b) has no multiple roots.

The roots of may be either real or complex. Then the expansion into partial fractions of has the form

where are scalars determined from and .

There is a simple way to compute the constant . Define the degree polynomial Multiply both sides of (??) by and evaluate at to obtain

For example, compute the partial fraction expansion for In this example, , , and . The relevant polynomials are: It follows that

##### Partial Fractions with Complex Roots

Suppose that the denominator has complex conjugate roots and . When is a complex simple root of , then the partial fractions expansion of contains the two terms where . Together these two terms must be real-valued, and it follows that . Therefore the expansion is for some complex scalar . These terms combine as where . With inverse Laplace transforms in mind, we prefer to write this expression as

In the third part of Section ?? we saw how to compute inverse Laplace transforms of functions like those in (??).

#### Partial Fractions Using MATLAB

The MATLAB command residue can be used to determine partial fractions expansions. We begin by discussing how polynomials are defined in MATLAB. The polynomial is stored in MATLAB by the vector q = [ad a1 a0] consisting of the coefficients of in descending order. For instance, in MATLAB, the polynomial is identified with the vector .

Suppose that the two vectors p and q represent the two polynomials (of degree less than ) and (of degree ). Both the vector of roots and the vectors of scalars are determined using the command

To illustrate this command, find the partial fraction expansion of (which we computed previously in (??)) by typing MATLAB responds with Note that this result agrees with our previous calculation in (??).

Observe that the situation where the polynomial has complex roots is not excluded. Indeed, let and , and type to obtain the answer In particular, has the three roots and we have the expansion

##### The Return to Real Form in Partial Fractions

We can return to a representation in real numbers by combining the terms corresponding to complex conjugate roots. When is a complex simple root of , then the partial fractions expansion of contains the two terms for some complex scalar . These terms combine to give where , as in (??).

We can write a MATLAB m-file to perform the computations in (??) as follows: This m-file is accessed using the command where i is the index corresponding to the complex conjugate root of . For example, if we consider the expansion in (??), then we type yielding the answer This output corresponds to the expression Combining the second and third term on the right hand side leads to

##### Repeating a Calculation Using MATLAB

The partial fraction expansion of (??), that is, is found by typing We obtain Hence we have the expansion Now we use realform to return to a representation avoiding complex numbers. Type to obtain which corresponds to (??).

### Exercises

In Exercises ?? – ?? use partial fractions to find a function whose Laplace transform is the given function .

In Exercises ?? – ?? use the MATLAB command residue to compute the expansion into partial fractions of for the given polynomials and .