Factor Theorem Proof
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, and
- 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.
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