Factor Theorem

Note: Before this see Remainder Theorem

if we calculate f(c) and it was 0? which implies remainder is 0,
then (x-c) must be a factor of the polynomial!

Example: x²-3x-4

f(4) = (4)²-3(4)-4 = 16-12-4 = 0

so (x-4) must be a factor of x²-3x-4


The Factor Theorem:

When f(c)=0 then x-c is a factor of the polynomial

we can also write:

When x-c is a factor of the polynomial then f(c)=0

 Why Is This Useful?

Knowing that x-c is a factor is the same as knowing that c is a root (and vice versa).

The factor "x-c" and the root "c" are the same thing

Example:

 2x³-x²-7x+2

The polynomial is degree 3, and could be difficult to solve. So let us plot it first:

The curve crosses the x-axis at three points, and one of them might be at 2. We can check easily:
f(2) = 2(2)³-(2)²-7(2)+2 = 16-4-14+2 = 0
Yes! f(2)=0, so we have found a root and a factor.

So (x-2) must be a factor of 2x³-x²-7x+2

How about where it crosses near -1.8?
f(-1.8) = 2(-1.8)³-(-1.8)²-7(-1.8)+2 = -11.664-3.24+12.6+2 = -0.304
No. (x+1.8) is not a factor.