What is factor theorem

If f(x) is a polynomial of degree n ≥ 1 and ‘a’ is any real number, then

• (x-a) is a factor of f(x) , if f(a)=0
• Its converse “ if (x-a) is a factor of the polynomial f(x), then f(a)=0”

In mathematics, factor theorem is used as a linking factor and zeros of the polynomial. Factor theorem is commonly used for factoring a polynomial and finding the roots of the polynomial equation.

Steps to Use Factor Theorem

Step 1 : If f(-c)=0, ( x+ c) is a factor of the polynomial f(x).

Step 2 : If p(d/c)= 0, (cx-d) is a factor of the polynomial f(x).

Step 3 : If p(-d/c)= 0, (cx+d) is a factor of the polynomial f(x).

Step 4 : If p(c)=0 and p(d) =0, then (x-c) and (x-d) is a factor of the polynomial.

Rather than finding the factors by using polynomial long division method, the best way to find the factors are factor theorem and synthetic division method. The factor theorem is mainly used to remove the known zeros from polynomials leaving all unknown zeros unimpaired, thus by finding the zeros easily to produce the lower degree polynomial.

Example:

Consider the polynomial function f(x)= x² +2x -15

The values of x for which f(x)=0 are called the roots of the function. By solving the equation, f(x)=0

Then, we get

x² +2x -15 =0

(x+5)(x-3)=0

(x+5)=0 or (x-3)=0

x = -5 or x = 3

Because (x+5) and (x-3) is a factor of x² +2x -15, -5 and 3 are the solutions to the equation x² +2x -15=0, we can also check as follows:

If x = -5 is the solution , then

f(x)= x² +2x -15

f(-5) = (-5)² + 2(-5) – 15

f(-5) = 25-10-15

f(-5)=25-25

f(-5)=0

If x=3 is the solution, them

f(x)= x² +2x -15

f(3)= 3² +2(3) – 15

f(3) = 9 +6 -15

f(3) = 15-15

f(3)= 0

If the remainder is zero, (x-c) is a polynomial of f(x)