Question

Can someone explain with examples that What is Factor Theorem???with Examples...

Answer 1

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

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)