1. Factor
Theorem

2. C2: Algebra Factor theorem
KUS objectives BAT perform algebraic long division BAT factorise cubic functions and other expressions BAT recall and use the Factor Theorem
Starter: Expand

3. Consider the function f(x) = x2 + 3x - 10 Factorise it:
BAT perform algebraic long division BAT factorise cubic functions and other expressions BAT recall and use the Factor Theorem
Introduction:
Consider the function f(x) = x2 + 3x - 10
Factorise it:
Then:
You can now sketch the graph of f(x).
The Factor Theorem states that if f(a) = 0 for a polynomial f(x)
then (x – a) is a factor of the polynomial f(x).
f(x) = (x + 5)(x – 2)
f(-5) = 0 and f(2) = 0

4. Notes 1
If f(p) = 0, then (x – p) is a factor of f(x)
f(x)  A function of x, any equation
f(p)  The function of x with a value p substituted in
For example;
Show that (x – 2) is a factor of x3 + x2 – 4x - 4
x3 + x2 – 4x - 4
Substitute in x = 2
– (4x2) - 4
Work out each term –
= 0
So because f(2) = 0, (x – 2) is a factor of the original equation

5. (x – 3) is a factor of 𝑓 𝑥 = 𝑥 3 −3 𝑥 2 −9𝑥+27
Notes 2
The Factor Theorem states that if f(a) = 0 for a polynomial f(x)
then (x – a) is a factor of the polynomial f(x).
for example
(x – 3) is a factor of 𝑓 𝑥 = 𝑥 3 −3 𝑥 2 −9𝑥+27
𝑓 3 =
(3) 3 − − =0
Since 𝑥 3 −3 𝑥 2 −9𝑥+27= 𝑥−3 𝑥 2 −9
When x = 3 the bracket = 0
so f(x) = 0

6. WB 6a
If f(p) = 0, then (x – p) is a factor of f(x)
f(x)  A function of x, any equation
f(p)  The function of x with a value p substituted in
For example;
Factorise 2x3 + x2 – 18x - 9
2x3 + x2 – 18x - 9
Substitute in values of x to find a factor
x = 1 –
= -24
x = 2 –
= -25
x = 3 –
So (x – 3) is a factor
= 0

7. If f(p) = 0, then (x – p) is a factor of f(x)
WB 6b
If f(p) = 0, then (x – p) is a factor of f(x)
f(x)  A function of x, any equation
f(p)  The function of x with a value p substituted in
For example;
Factorise 2x3 + x2 – 18x – 9
 Now we know (x – 3) is a factor, divide by it to find the quotient
The quotient is 2x2 + 7x + 3
2x2 + 7x + 3
x - 3
2x x2 – 18x - 9
2x3 – 6x2
7x2 – 18x - 9
Third, divide 3x by x
= 3
Then, work out 3(x – 3) and subtract from what you have left
Second, divide 7x2 by x
= 7x
Then, work out 7x(x – 3) and subtract from what you have left
First, divide 2x3 by x
= 2x2
Then, work out 2x2(x – 3) and subtract from what you started with
7x2 – 21x
3x - 9
3x - 9

8. If f(p) = 0, then (x – p) is a factor of f(x)
WB 6c
If f(p) = 0, then (x – p) is a factor of f(x)
f(x)  A function of x, any equation
f(p)  The function of x with a value p substituted in
For example;
Factorise 2x3 + x2 – 18x – 9
given that f(3) = 0
 (x – 3) is a factor
 (2x2 + 7x + 3) is the quotient
(x – 3)(2x2 + 7x + 3)
You can also factorise the quotient
2 numbers that multiply to give +3, and add to give +7 when one has doubled…
(x – 3)(2x + 1)(x + 3)

9. b) Find a given that (𝑥−2) is a factor of 𝑥 3 + 𝑎𝑥 2 – 4𝑥+6.
BAT perform algebraic long division BAT factorise cubic functions and other expressions BAT recall and use the Factor Theorem
WB 7
a) Given that (x + 1) is a factor of 4x4 – 3x2 + a, find the value of a.
4x4 – 3x2 + a
If (x + 1) is a factor, then using -1 will make the equation = 0
0 =
4(-14) – 3(-12) + a
0 = 4 – a
Work out each term
0 = a
Solve the equation to find the value of a
-1 = a
b) Find a given that (𝑥−2) is a factor of 𝑥 3 + 𝑎𝑥 2 – 4𝑥+6.
(2)3 + a(2)2 - 4(2) + 6 = 0
4a + 6 = 0
𝒂 =− 𝟑 𝟐

10. So 𝑥 3 −2 𝑥 2 −8𝑥=(𝑥+2)( 𝑥 2 −4𝑥) 𝑥 2 −4𝑥 𝑥 +2 𝑥 3 −4𝑥 2 +2𝑥 2 −8𝑥
Algebra division: Divide by a bracket using table of values
Divided by
𝑥 2
−4𝑥 𝑥 +2
𝑥 3
−4𝑥 2
+2𝑥 2
−8𝑥
𝑟𝑒𝑚𝑎𝑖𝑛𝑑𝑒𝑟 0
Use EWB pen to complete the example
So 𝑥 3 −2 𝑥 2 −8𝑥=(𝑥+2)( 𝑥 2 −4𝑥)

11. So 𝑥 3 +9 𝑥 2 +26𝑥+24=(𝑥+3)( 𝑥 2 +6𝑥+8) 𝑥 2 +6𝑥 +8 𝑥 +3 𝑥 3 +6𝑥 2 +8𝑥
Algebra division: Divide by a bracket using table of values
Divided by
𝑥 2
+6𝑥 +8 𝑥 +3
𝑥 3
+6𝑥 2
+8𝑥
+3𝑥 2
+18𝑥
+24
𝑟𝑒𝑚𝑎𝑖𝑛𝑑𝑒𝑟 0
Use EWB pen to complete the example
So 𝑥 3 +9 𝑥 2 +26𝑥+24=(𝑥+3)( 𝑥 2 +6𝑥+8)

12. 1. Show that (x – 2) is a factor of x3 - 4x2 + x + 6.
BAT perform algebraic long division BAT factorise cubic functions and other expressions BAT recall and use the Factor Theorem
Practice 1
1. Show that (x – 2) is a factor of x3 - 4x2 + x + 6.
Hence factorise the expression completely.
2. Factorise the expression x3 - 3x2 - 10x + 24.
Hence solve the equation x3 - 3x2 - 10x + 24 = 0.
3. If (x – 2) is a factor of f(x) = x3 - 3x2 + a, find the value of a.
4. Determine whether or not (x – 3) is a factor of the
expression, x3 - 6x2 + 5x + 12.

13. Use the factor theorem to show that the following are factors, and hence fully factorise
Practice 2
easy
harder
is a factor of
is a factor of
is a factor of
is a factor of
is a factor of
is a factor of
is a factor of
is a factor of
is a factor of
is a factor of

14. Check your answer using Geogebra
BAT perform algebraic long division BAT factorise cubic functions and other expressions BAT recall and use the Factor Theorem
WB 8
Factorise the cubic polynomial f(x) = x3 – 2x2 – x + 2 and
hence sketch the graph of the function.
Check your answer using Geogebra

15. Check your answer using Geogebra
BAT perform algebraic long division BAT factorise cubic functions and other expressions BAT recall and use the Factor Theorem
WB 9
Express f(x) = x3 + x2 – 5x + 3 as the product of three
linear factors. Hence:
a) Sketch the graph of the function.
b) Solve the equation x3 + x2 – 5x + 3 = 0
Check your answer using Geogebra

16. Check your answer using Geogebra
BAT perform algebraic long division BAT factorise cubic functions and other expressions BAT recall and use the Factor Theorem
WB 10: Factorise the cubic polynomial f(x) = x3 + 3x2 – 12x – 14
Hence:
a) Sketch the graph of the function.
b) Solve f(x) = 0
Check your answer using Geogebra

17. The polynomial 𝑝(𝑥)is given by 𝑝 𝑥 = 𝑥 3 −19𝑥−30
BAT perform algebraic long division BAT factorise cubic functions and other expressions BAT recall and use the Factor Theorem
WB11 exam question
The polynomial 𝑝(𝑥)is given by 𝑝 𝑥 = 𝑥 3 −19𝑥−30
Use the factor theorem to show that 𝑥+2 is a factor of 𝑝(𝑥)
Express 𝑝 𝑥 as the product of three linear factors

18. factor theorem challenge

20. One thing to improve is –
KUS objectives BAT perform algebraic long division BAT factorise cubic functions and other expressions BAT recall and use the Factor Theorem
self-assess
One thing learned is –
One thing to improve is –

21. END