Introduction to factor theorem — lesson. Mathematics CBSE, Class 9.

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Factor Theorem: If

$p (x)$ is a polynomial of degree

$n > 1$ and

$a$ is any real number, then:

(i)

$x - a$ is a factor of

$p (x)$ , if

$p (a) = 0$ , and

(ii)

$p (a) = 0$ , if is a factor of

$p (x)$ .

This is an extension to the remainder theorem where the remainder is

$0$ , which is

$p (a) = 0$ .

Example:

Examine whether

$x + 2$ is factor of

$p (x)= x^{3} + 3 x^{2} + 5 x + 6$ .

To find

$x + 2$ is a factor of

$p (x)= x^{3} + 3 x^{2} + 5 x + 6$ or not.

As, result of factor theorem

$x + 2$ is factor of

$p(x)$ if

$p (- 2) = 0$ .

The zero of

$x + 2$ is

$x+2 = 0$ . That is,

$x = -2$ .

$\begin{array}{l} p (- 2) = {(- 2)}^{3} + 3 {(- 2)}^{2} + 5 (- 2) + 6 \\ p (- 2) = - 8 + 12 - 10 + 6 \\ p (- 2) = - 18 + 18 \\ p (- 2) = 0 \end{array}$

Therefore,

$x + 2$ is factor of

$p (x)= x^{3} + 3 x^{2} + 5 x + 6$ .

Important!

When

$x - a$ may be a factor of

$p (x)$ then

$p (a) = 0$ .

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1. Introduction to factor theorem

Theory: