Factor Theorem with Examples and FAQs

Factor Theorem

The factor theorem is used for factorizing a polynomial, which helps to determine the roots of the polynomial. Factor theorem is known to be a special case of a polynomial remainder theorem.

Statement:

let f(x) be a polynomial of  degree n>1 and a be any real number

Proof:

\text { (1) Let } f(a)=0 \text {. }

On the division of f(x) \text { by }(x-a), we will get a quotient.

Let the quotient be q(x).

By The Remainder Theorem, when fx is divided by x-a, then the remainder is  f(a).

\therefore f(x)=(x-a) \cdot q(x)+f(a)

(2) Let x-a is a factor of f(x).

On the division of f(x)by (x-a), we will get a quotient. Let the quotient be q(x).

\therefore f(x)=(x-a) \cdot q(x)

Putting x = a  in the above equation we get, 

f(a)=(a-a) \cdot q(a) \Rightarrow f(a)=0 \cdot q(x)

Thus, x-a is a factor of f(x) \Rightarrow f(a)=0

For example, Consider the polynomial function f(x)=x^{2}-17 x+16

Let’s find the values of x for which f(x)=0.

Solve the equation, assuming f(x)=0

\Rightarrow x^{2}-17 x+16=0

∴x-16 and x-1 are factors of fx and x=16 or  x=1 are the solution to the equation x^{2}-17 x+16=0

Now to do the converse, and check if x-16 and x-1 are factors of fx, we will put the value of x as x=16 and  x=1, one by one in the polynomial function, f(x)=x^{2}-17 x+16

For x=16,

f(16)=16^{2}-17(16)+16=256-272+16=0

\Rightarrow f(16)=0

For x=1,

f(1)=1^{2}-17(1)+16=1-17+16=0

\Rightarrow f(1)=0

Thus, x-16 and x-1 are factors of f(x).

Example

Use the factor theorem to check if (x-5) is a factor of f(x)=3 x^{2}-8 x-35

Solution

x-5=0⇒x=5

By factor theorem, (x-5) will be a factor of fx=3x2-8x-35, if f(x)=3 x^{2}-8 x-35, \text { if } f(5)=0.

f(x)=3 x^{2}-8 x-35

f(5)=3(5)^{2}-8(5)-35=75-40-35=0

\Rightarrow(x-5) \text { is a factor of } f(x)=3 x^{2}-8 x-35

Hence (x-5) is a factor of the given polynomial f(x)