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Factor theorem is used for finding the roots of the given polynomial. This theorem is very helpful in finding the factors of the polynomial equation without actually solving them.

According to the factor theorem, for any polynomial f(x) of degree n ≥ 1 a linear polynomial (x – a) is the factor of the polynomial if f(a) is zero.

Let’s learn about the factor theorem, its proof, and others in detail in this article.

What is Factor Theorem?

A special theorem that links polynomials with their zeros and helps to find the factors of the polynomial is called the Factor Theorem. Factor theorem along with the remainder theorem is very helpful in solving complex polynomial equations. For any polynomial of a higher degree factor, the theorem removes all the known zeros and reduces the polynomial to a lesser degree, and then its factors are easily calculated. 

Factor Theorem Statement

According to the factor theorem, any polynomial f(x) of degree greater than or equal to 1 its factor can be easily found if we know the root of the polynomial equation. The root of the polynomial equation is the number that satisfies the polynomial equation. Suppose the root of f(x) is x = a then according to the factor theorem (x- a) is the factor of the polynomial f(x).

The image given below shows the factor theorem statement.

Factor Theorem Statement

 

Factor Theorem Formula

According to the factor theorem x – a can be considered the factor of the polynomial f(x) if f(a) is zero. Now the Factor Theorem Formula is,

If (x – a) is a factor of f(x) then,

f(a) = 0, 

Now f(x) can be expanded as,

f(x) = (x-a)q(x)

where,
a is the root of the equation
q(x) is the quotient when f(x) is divided by q(a)

Zero of a Polynomial

The concept of zero of the polynomial is very useful for understanding the factor theorem. Zero of any Polynomial is the real number that satisfies the polynomial equation. i.e. For polynomial f(x) ‘a’ is the zero of f(x) if f(a) is zero.

For example, the zero of the polynomial x2 + 4x – 12 is calculated as,

Solution:

x2 + 4x – 12

=  x2 + 6x -2x – 12

= x(x + 6) – 2(x + 6)

= (x -2)(x+6)

here,

zeros of the polynomial x2 + 4x – 12 are,

x – 2 = 0

x = 2

x + 6 = 0

 x = -6

Verification: To verify the zero of the polynomial x2 + 4x – 12

f(2) = 22 + 4(2) -12

      = 4 + 8 -12

      = 0

f(-6) = (-6)2 + 4(-6) – 12

        = 36 – 24 -12

        = 0

Thus, zero of the polynomial x2 + 4x – 12 is verified.

Proof of Factor Theorem

Consider a polynomial p(x) that is being divided by (x – b) only if p(b) = 0. 

Given Polynomial can be written as,

Dividend = (Divisor × Quotient) + Remainder

By using the division algorithm,

p(x) = (x – b) q(x) + remainder

where,
p(x) is the dividend
(x – b) is the divisor
q(x) is the quotient

From the remainder theorem, 

p(x) = (x – b) q(x) + p(b).

Suppose that p(b) =0.

p(x) = (p – b) q(x) + 0

p(x) = (x – b) q(x)

Thus, we can say that (x – b) is a factor of the polynomial p(x). 

Here we can see that the factor theorem is actually a result of the remainder theorem, which states that a polynomial (x) has a factor (x – a), if and only if, a is a root i.e., p(b) = 0.

How to Use the Factor Theorem?

Factor Theorem is used widely in solving the polynomial equation. We can learn about the use of factor theorem by going through the example discussed below,

Example: Find the factor of x2 – 4x + 3.

Solution:

Given polynomial,

f(x) =  x2 – 4x + 3

by hit and try method,

taking x = 1

f(1) = 12 -4(1) + 3

      = 0

Thus, x -1 is the factor of x2 – 4x + 3

again taking x = 3

f(3) = 32 – 4(3) + 3

      = 0

Thus, x -3 is the factor of x2 – 4x + 3

Using the Factor Theorem To Factor a Cubic Polynomial

Factor Theorem To Factor a Third-Degree Polynomial is widely used for solving the three-degree polynomial also called a cubic polynomial. Generally for quadratic polynomial factorization method is used and the factor theorem is used when we have to find the factors of any cubic polynomial. 

Use the steps given below to find the Factor of a Cubic Polynomial

Factorize the cubic polynomial f(x)  = ax3 + bx2 + cx + d.

Step 1: By hit and try method find one of the zeros of the polynomial f(x). Say the zero of the polynomial is “a” such that f(a) is zero.

Step 2: Divide the polynomial f(x) by x-a using synthetic division of the polynomial method.

Step 3: Using the division algorithm, write the given cubic polynomial as the f(x) = (x-a)g(x), where g(x) is a quadratic polynomial.

Step 4: Factorize the cubic polynomial g(x). As, g(x) = (x-b)(x-c) for any real numbers b and c.

Step 5: Express the given cubic polynomial as the product of these factors.

f(x) = (x -a)(x-b)(x-c)

Example Using factor theorem factorize f(x) = x3 + 10x2 + 23x + 14

Solution:

f(x) = x3 + 10x2 + 23x + 14

Now applying Factor Theorem.

(x – a) is a factor of f(x) if f(a) = 0.

And g(y) = (y-a)q(a)

Now, by hit and try method we get,

f(-1) = (-1)3 + 10(-1)2 + 23(-1) + 14

= 24 – 24

= 0

Thus, x + 1 is a factor of  x3 + 10x2 + 23x + 14

Now, by dividing x + 1 by  x3 + 10x2 + 23x + 14 we get,

Synthetic division of x3 + 10x2 + 23x + 14 by x + 1

 

f(x) = (x + 1)(x + 9x + 14)

      = (x + 1)(x + 7x + 2x + 14)

      = (x + 1){x(x +7) + 2(x + 2)}

f(x) = (x + 1)(x + 2)(x + 7)

Difference Between Factor Theorem and Remainder Theorem

Remainder Theorem and Factor Theorem are the basic theorems that help in mathematics and the basic difference between them is stated in the table given below,

Remainder TheoremFactor Theorem
According to, Remainder Theorem for any polynomial p(x) when divided by x – a the remainder is p(a).According to Factor Theorem if (x – a) is a factor of p(x) then this is true only if f(a) = 0.
Remainder Theorem helps us to find the remainder of the polynomial without actually dividing it.Factor theorem helps us to find the factor of the given polynomial.

For example, for the given polynomial f(x) = x2 + 3x – 4 the remainder when f(x) is divided by x – 2 = 0 is,

Solution:

Given,

x – 2 = 0
x = 2

f(2) = 22 + 3(2) – 4
      = 6

Hence, the remainder when we divide x2 + 3x – 4  by x – 2 is 6.

Similarly, for the given polynomial f(x) = x2 + 3x – 4 the remainder when f(x) is divided by x – 2 = 0 is “0” then we can say that x -2 is a factor of x2 + 3x – 4

Given,

x – 1 = 0
x = 1

f(2) = 12 + 3(1) – 4
      = 0

Hence, the remainder when we divide x2 + 3x – 4  by x – 2 is 0.

Thus, according to the factor theorem x -2 is a factor of x2 + 3x – 4.

Read, More

Solved Examples on Factor Theorem

Example 1: Find the root of the polynomial f(x) = x2 – 5x + 6 using the factor theorem.

Solution:

f(x) = x2 – 5x + 6.

Now applying factor theorem.

If (x – a) is a factor of f(x) then f(a) = 0

By hit and try method,

f(2) = 4 – 10 + 6 

      = 0

Thus, (x – 2) is a factor of f(x).

Example 2: Suppose a polynomial f(x) = x3 + 3x2 – 6x – 18. If x = -3 is a root show that x + 3 is a factor.

Solution:

f(x) = x3 + 3x2 – 6x – 18.

Now applying factor theorem.

If x = a is the root of f(x) then x – a is the factor of the f(x)

Now for, f(x) at x = -3

f(-3) = -27 + 27 + 18 – 18 
        = 0

Thus, (x + 3) is a factor of f(x).

Example 3: Using the factor theorem check if y + 2 is a factor of the polynomial 2y4 + y3 – 2y2 + 4y -8, or not.

Solution:

Given y + 2

If, y + 2 = 0, then y = -2

Substituting y = -2 in the given polynomial 2y4 + y3 – 2y2 + 4y -8

= 2(-2)4 + (-2)3 – 2(-2)2 + 4(-2) – 8

= 32 – 8 – 8 – 8 – 8

= 0

Thus, we can say that y + 2 is a factor of the polynomial 2y4 + y3 – 2y2 + 4y -8

Example 4: Check if x – 1 is a factor of the polynomial f(x) = 2x4 + 3x2 – 5x + 7

Solution:

Given x – 1

If, x – 1 = 0, then x = 1

Substituting x = 1 in the given polynomial 2x4 + 3x2 – 5x + 7

= 2(1)4 + 3(1)2 – 5(1) + 7

= 2 + 3 – 5 + 7

= 7

As, f(-1) ≠ 0

We can say that x – 1 is not a factor of the polynomial  2x4 + 3x2 – 5x + 7

Example 5: Given a polynomial f(x) = x3 – x 2 – 2x + 1. Factorize and find its roots.

Solution:

f(x) = x3 – x 2 – x + 1

Now applying Factor Theorem.

(x – a) is a factor of f(x) if f(a) = 0.

And g(y) = (y-a)q(a)

Now, by hit and try method we get,

f(-1) = (-1)3 – (-1)2 -(-1)

= -8 + 4 + 4

= 0

Thus, x + 1 is a factor of x3 – x 2 – x + 1

Now, by dividing x + 1 byx3 – x 2 – x + 1 we get,

f(x) = (x + 1)(x2 – 2x + 2)

= (x + 1)(x – x – x -2)

= (x + 1){x(x – 1) – 1(x – 1)}

f(x) = (x + 1)(x – 1)(x – 1)

FAQs on Factor Theorem

Q1: What is Factor Theorem?

Answer:

Factor theorem is one of the basic theorems in mathematics it states that for any polynomial p(x) whose degree is greater than equal to 1 x – a is a factor for any real number “a” if p(a) is zero.

Q2: What is the Factor Theorem Formula?

Answer:

The Factor Theorem Formula is discussed below,

For any polynomial f(x) if it has (x – a) as its factor then it is true only when,

f(a) = 0, 

Now f(x) can be expanded as,

f(x) = (x-a)q(x)

where,
a is the root of the equation
q(x) is the quotient when f(x) is divided by q(a)

Q3: What is the Importance of the Factor Theorem?

Answer:

Factor theorem is widely used for finding the roots of the polynomial function and it is also used to factorize polynomial equations.

Q4: How to Use the Factor Theorem?

Answer:

The factor theorem is widely used in Mathematics it is also used for various real-life purposes.  For example, is used for dividing and comparing various quantities, calculating estimated values and others.



Last Updated : 18 Apr, 2023
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