Zero of a Polynomial
The concept of zero of the polynomials 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)
zeros of the polynomial x2 + 4x – 12 are,
x – 2 = 0
⇒ x = 2, and
x + 6 = 0
⇒ x = -6
Verification: To verify the zero of the polynomial x2 + 4x – 12
f(2) = 22 + 4(2) -12
⇒ f(2) = 4 + 8 -12 = 0
f(-6) = (-6)2 + 4(-6) – 12
⇒ f(-6) = 36 – 24 -12 = 0
Thus, zero of the polynomial x2 + 4x – 12 is verified.
Factor Theorem
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.
Table of Content
- What is the Factor Theorem?
- Factor Theorem Statement
- Factor Theorem Formula
- Zero of a Polynomial
- Factor Theorem Proof
- How to Use Factor Theorem?
- Using the Factor Theorem To Factor a Cubic Polynomial
- Factor Theorem and Remainder Theorem
- Factor Theorem Examples