** Definition of Factor Theorem**

- According to Factor Theorem, if
*s*is the root of any polynomial P(*x*), i.e. if P(*s*) = 0, then (*x*-*s*) is the factor of the polynomial P(*x*).

**Example of Factor Theorem**

- To factor
*x*^{3}+ 3*x*^{2}+ 3*x*+ 1, first find the zero of the polynomial.

Let f(*x*) =*x*^{3}+ 3*x*^{2}+ 3*x*+ 1

Let us check*x*= 1 by trial and error method, then,*f*(1) = (1)^{3}+ 3(1)^{2}+ 3(1) + 1 = 1 + 3 + 3 + 1 = 8 which is not equal to zero.

So, (*x*– 1) is not a factor.

Now check*x*= - 1,*f*(- 1) = (- 1)^{3}+ 3(- 1)^{2}+ 3(- 1) + 1 = - 1 + 3 - 3 + 1 = 0.

Therefore, (*x*+ 1) is a factor of the given polynomial.

By dividing the given polynomial with (*x*+ 1), we get (*x*^{2}+ 2*x*+ 1).

Factoring this trinomial, we get (*x*+ 1) (*x*+ 1).

So, the factors of the given polynomial are (*x*+ 1), (*x*+ 1), and (*x*+ 1).

**Solved Example on Factor Theorem**

Identify the factors of the polynomial 2

x^{2}- 5x+ 3.

Choices:

A. (2x- 3)(x- 1)(x+ 1)

B. (2x- 3)(x- 1)

C. (2x- 3)(x- 2)(x+ 1)

D. (2x- 3)(x+ 1)(x+ 2)

Correct Answer: B

Solution:

Step 1:Letf(x) = 2x^{2}- 5x+ 3

Step 2:Let us checkx= 1 by trial and error method. Then,f(1) = 2(1)^{2}- 5(1) + 3 = 2 - 5 + 3 = 0

Step 3:According to factor theorem, iff(1) = 0, then (x- 1) is a factor of the polynomialf(x).

Step 4:Therefore, (x- 1) is a factor of the given polynomial.

Step 5:By dividing the given polynomial with (x- 1), we get (2x- 3)

Step 6:So, the factors of the given polynomial are (2x- 3) and (x- 1).

**Related Terms for Factor Theorem**

- Factor
- Polynomial