1. The Factor Theorem

2. What does it do? It helps you factorise polynomials of high order…..
like 3!

3. Function notation, f(x)
f(x) stands for “a function of x”
This basically says “a polynomial using the letter x”
So f(x) could be:
f(x) = x3 + 2x2 – x – 2
It saves you having to write out the polynomial every time you refer to it.
Also……

4. Function notation
We can use this notation when we substitute values into the polynomial:
f(x) = x3 + 2x2 – x – 2
f(1) is just what you get when you substitute 1 into the polynomial
f(-2) is what you get when you substitute -2 into the polynomial
What is f(3)?

5. Let’s start simple Factorise x2 – 5x – 6 (x – 6)(x + 1)
Now solve x2 – 5x – 6 = 0
(x – 6)(x + 1) = 0
So either x – 6 = 0 or x + 1 = 0
x = x = -1

6. f(x) = x2 – 5x – 6
What are the values of f(6) and f(-1) (i.e. what do you get when you substitute the values 6 and -1 into the polynomial?)
Why?

7. A little bit harder… x3 + 2x2 – x – 2 What is f(-2) ?
What happens when you divide this polynomial by (x + 2)?

8. Can we see a pattern yet? x3 + 2x2 – x – 2
What happens when you divide this polynomial by (x – 1)?
What do you think the value of f(1) will be?

9. The Factor Theorem If (x – a) is a factor of a polynomial f(x), then:
x = a is a solution (root) of the equation f(x) = 0.
Conversely, if f(a) = 0, then (x – a) is a factor of f(x)

10. Using the theorem f(x) = x3 – x2 – 4x + 4
What are f(-2), f(-1), f(0), f(1), f(2)?
What are the factors of f(x)?