How Do You Use The Factor Theorem

Factor Theorem

Theorem: If p(x) is a polynomial of degree n ≥ 1 and a is any real number, then
(i) x – a is a factor of p(x), if p(a) = 0, and (ii) p(a) = 0, if x – a is a factor of p(x).
Proof: By the Remainder Theorem,
p(x) = (x – a) q(x) + p(a).
(i) If p(a) = 0, then p(x) = (x – a) q(x),
which shows that x – a is a factor of p(x).
(ii) Since x – a is a factor of p(x),
p(x) = (x – a) g(x) for same polynomial g(x).
In this case, p(a) = (a – a) g(a) = 0.

To use factor theorem

  • Step 1: (x + a) is factor of a polynomial p(x) if p(–a) = 0.
  • Step 2: (ax – b) is a factor of a polynomial p(x) if p(b/a) = 0
  • Step 3: ax + b is a factor of a polynomial p(x) if p(–b/a) = 0.
  • Step 4: (x – a) (x – b) is a factor of a polynomial p(x) if p(a) = 0 and p(b) = 0.

Factor Theorem Example Problems With Solutions

Example 1:    Examine whether x + 2 is a factor of x3 + 3x2 + 5x + 6 and of 2x + 4.
Solution:    The zero of x + 2 is –2.
Let p(x) = x3 + 3x2 + 5x + 6 and s(x) = 2x + 4
Then,    p(–2) = (–2)3 + 3(–2)2 + 5(–2) + 6
= –8 + 12 – 10 + 6
= 0