The Factor Theorem is a special case of the Remainder Theorem. It is a rule in algebra used to check whether a given expression is a factor of a polynomial.
Statement: If f(a) =0, then (x - a) is a factor of f(x).
Solved Examples
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) = x + 3x2 - 6x - 18. If x = -3 is a root, then 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 - 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)
⇒ f(-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)
⇒ f(x) = (x + 1)(x2 - x - x -2)
⇒ f(x) = (x + 1){x(x - 1) - 1(x - 1)}
⇒ f(x) = (x + 1)(x - 1)(x - 1)
Practice Question
Question 1: Using the Factor Theorem, factorize f(x) = x3 + 8x2 + 17x + 10
Question 2: Using the Factor Theorem, factorize f(x) = x3 + 3x2 − 4x − 12
Question 3: Find all the factors of f(x) = 2x3 + x2 − 8x − 4 using the Factor Theorem.
Question 4: Verify a suitable linear factor and factorize f(x) = 2x3 − x2 − 18x + 9 completely using the Factor Theorem.
Question 5: Determine the linear factors of f(x) = 2x3 − 3x2 − 11x + 6 using the Factor Theorem.