explain factor theorem with example
Answers
When f(x) is divided by (x – a), we get
f(x) = (x – a)Q(x) + remainder
From the Remainder Theorem, we get
f(x) = (x – a)Q(x) + f(a)
If f(a) = 0 then the remainder is 0 and
f(x) = (x – a)Q(x)
We can then say that (x – a) is a factor of f(x)
The Factor Theorem states that
(x – a) is a factor of the polynomial f(x) if and only if f(a) = 0
Take note that the following statements are equivalent for any polynomial f(x).
• (x – a) is a factor of f(x).
• The remainder is zero when f(x) is divided by (x – a).
• f(a) = 0.
• The solution to f(x) = 0 is a.
• The zero of the function f(x) is a.
Example:
Determine whether x + 1 is a factor of the following polynomials.
a) 3x4 + x3 – x2 + 3x + 2
b) x6 + 2x(x – 1) – 4
Solution:
a) Let f(x) = 3x4 + x3 – x2 + 3x + 2
f(–1) = 3(–1)4 + (–1)3 – (–1)2 +3(–1) + 2
= 3(1) + (–1) – 1 – 3 + 2 = 0
Therefore, x + 1 is a factor of f(x)
b) Let g(x) = x6 + 2x(x – 1) – 4
g(–1) = (–1)6 + 2(–1)( –2) –4 = 1
Therefore, x + 1 is not a factor of g(x)