1. “Teach A Level Maths” Vol. 1: AS Core Modules
21:The Factor Theorem
© Christine Crisp

2. A simple quadratic function can be factorised by inspection e. g
A simple quadratic function can be factorised by inspection e.g. Consider
The factors are
From the factors, we can see that the roots of the function are
and
So,
Reversing this process enables us to factorise cubics ( and polynomials of higher degrees )

3. e.g. If
is a factor
This time there is no remainder
For a polynomial function , the factor theorem says that:
if then is a factor

4. Factorising a Cubic Function
e.g.1 Find a linear factor of
Solution: Let
Try x = 1:
is a factor
Once we have found one factor of a cubic, the other factor, which is quadratic, can be found by long division.
Sometimes this quadratic factor will also factorise

5. e.g.2 Factorise fully
Solution: We use the factor theorem to find one linear factor:
Let
is not a factor
is a factor
So,
quadratic factor
The quadratic factor can be found by long division.

6. The quadratic factor has no real roots as b2–4ac<0
x3+ x2
-3x2+ 5x
-3x2– 3x
8x+8
8x+8
We have
The quadratic factor has no real roots as b2–4ac<0

7. 1 real root at x = –1

8. SUMMARY Factorising Cubic Functions
Use the factor theorem to find one linear factor
if then is a factor
Use long division to find the quadratic factor
Factorise the quadratic factor if possible

9. Exercises
Factorise the following cubics: 1. 2. 3.

11. The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied.
For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.

12. SUMMARY Factorising Cubic Functions
Use the factor theorem to find one linear factor
Use inspection to find the quadratic factor
Start with the term of the cubic
Find the constant
Use the term of the cubic to find the middle term of the quadratic factor
Factorise the quadratic factor if possible
Check factors using the x term of the cubic
Factorising Cubic Functions
if then is a factor

13. Solution: We use the factor theorem to find one linear factor:
e.g. Factorise fully
Let
We could try any of
is not a factor
is a factor
So,
quadratic factor
The quadratic factor can be found by inspection.
In this example the quadratic factor has no linear factors.