Factor Theorem

In mathematics, factor theorem is used when factoring the polynomials completely. It is a theorem that links factors and zeros of the polynomial.

According to 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.

Also, we can say, if (x-a) is a factor of polynomial f(x), then f(a) = 0. This proves the converse of the theorem. Let us see the proof of this theorem along with examples.

What is a Factor Theorem?

Factor theorem is commonly used for factoring a polynomial and finding the roots of the polynomial. It is a special case of a polynomial remainder theorem.

As discussed in the introduction, a polynomial f(x) has a factor (x-a), if and only if, f(a) = 0. It is one of the methods to do the factorisation of a polynomial.

Proof

Here we will prove the factor theorem, according to which we can factorise the polynomial.

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)

Where f(x) is the target polynomial and q(x) is the quotient polynomial.

Since, f(c) = 0, hence,

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).

Another Method

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

How to Use Factor Theorem 

The steps are given below to find the factors of a polynomial using factor theorem:

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

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

Step 3 : If p(-d/c)= 0, then (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) are factors of the polynomial p(x).

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 used primarily 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.

Other Methods to Find Factors

Apart from factor theorem, there are other methods to find the factors, such as:

• Polynomial Long Division
• Synthetic Division

Problems 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.

Solving the equation, assume f(x)=0, we get:

x² +2x -15 =0

x² +5x – 3x -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) are factors of x² +2x -15, -5 and 3 are the solutions to the equation x² +2x -15=0, we can also check these 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 the synthetic division method to find the remainder.

Consider the same polynomial equation

f(x)= x² +2x -15

We use 3 on the left in the 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.

Keep visiting BYJU’S for more information on polynomials and try to solve factor theorem questions from worksheets and also watch the videos to clarify the doubts.