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Factor Theorem

In mathematics, factor theorem is used as a linking factor and zeros of the polynomial.

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”

Factor Theorem Proof

Consider a polynomial f(x) which is divided by (x-c), then f(c)=0

Using remainder theorem,

f(x)= (x-c)q(x)+f(c)

f(x) = (x-c)q(x)+0

f(x) = (x-c)q(x)

Therefore, (x-c) is a factor of the polynomial f(x)

Proof by Remainder Theorem

By remainder theorem,

f(x)= (x-c)q(x)+f(c)

If (x-c) is a factor of f(x), then the remainder must be zero

(x-c) exactly divides f(x)

Therefore, f(c)=0.

The following statements are equivalent for any polynomial f(x)

• The remainder is zero, when f(x) is divided by (x-c)
• (x-c) is a factor of f(x)
• C is the solution to f(x)=0
• C is a zero of the function f(x), or f(c) =0

Factor Theorem Procedure

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. This 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.

There is another way to define the factor theorem. Usually, when a polynomial is divided by a binomial, we will get a reminder. The quotient obtained is called as depressed polynomial when the polynomial is divided by one of its binomial factors. If you get the remainder as zero, the factor theorem is illustrated as follows:

The polynomial, say f(x) has a factor (x-c) if f(c)= 0, where f(x) is a polynomial of degree n, where n is greater than or equal to 1 for any real number, c.

Examples and Solutions

Factor theorem example and solution are given below. Go through once and get a clear understanding of this theorem. Factor theorem class 9 maths polynomial enables the children to get a knowledge of finding the roots of quadratic expressions and the polynomial equations which is used for solving complex problems in your higher studies.

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, then;

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)

Alternate Method – Synthetic Division Method

We can also use synthetic division method to find the remainder.

Consider the same polynomial equation

f(x)= x2 +2x -15

We use 3 on the left in synthetic division method along with the coefficients 1,2 and -15 from the given polynomial equation.

Since the remainder is zero, 3 is the root or solution of the given polynomial.

The techniques used for solving the polynomial equation of degree 3 or higher are not as straightforward. So linear and quadratic equations are used to solve the polynomial equation.