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Rest theorem


Let's calculate the rest of the division of :

R (x) = 3

The root of the divider is .
Note that:

That is when B (x) is a polynomial of degree 1, the rest is equal to the numerical value of P (x) When x assumes the root value of B (x).

To demonstrate this fact, let's effect:

Note that the degree of rest is 0because it is smaller than the divider degree, which is 1. So the rest is a constant r.

Performing we have:

Thus, we can state the following theorem:

Rest theorem

The rest of division of a polynomial P (x) by the binomial ax + b equals the numerical value of this polynomial for , that is, .

Example

Calculate the remainder of the division of P (x) = x² + 5x - 1 per B (x) = x + 1:

Resolution

We found the root of the divisor:

x + 1 = 0  x = - 1

From the rest theorem, we know that the rest is equal to P (-1):

P (-1) = (-1) ² + 5. (-1) -1  P (- 1) = - 5 = r

So the rest of the division of x² + 5x - 1 per x + 1 é - 5.

note that P (x) is divisible by ax + b When r = 0, ie when . Hence comes the statement of the following theorem:

D'Alembert's theorem

A polynomial P (x) is divisible by the binomial 1 if and only if .

The most important case of division of a polynomial P (x) is the one where the divisor is of the form (x - ).

note that is the root of the divisor. So the rest of the division of P (x) by (x -) é:

r = P ()

Like this:

P (x) is divisible by (x - ) When r = 0, ie when P() = 0.

Example

Determine the value of Pso that the polynomial be divisible by x - 2:

Resolution

For what P (x) be divisible by x - 2 we should have P (2) = 0, because 2 is the root of the divisor:

So for that be divisible by x - 2 we should have p = 19.

Next: Division of a Polynomial by (x-a) (x-b)