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

Did you find an error?

1. Introduction to factor theorem