The Factor Theorem
What does it do?
It helps you factorise polynomials of high
order…..
like 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……
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)?
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=6
x = -1
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?
A little bit harder…
x3 + 2x2 – x – 2
What is f(-2) ?
What happens when you divide this
polynomial by (x + 2)?
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?
The Factor Theorem
If (x – a) is a factor of a polynomial f(x),
then:
• f(a) = 0
• 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)
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)?