iGCSE (2021 Edition)

# 5.04 Factorise cubic polynomials

Worksheet
Factorising cubic polynomials
1

Factorise the following expressions:

a

x^{3} + 27

b

x^{3} + y^{3}

c

x^{3} - 64

d

x^{3} + 64 y^{3}

e

x^{3} - 125 y^{3}

f

x^{3} + 729 y^{3}

g

27 x^{3} + 64

h

512 x^{3} - 343

i

40 u^{3} + 5 v^{3}

j

4 m^{3} - 32 n^{3}

k

\left(x + 7\right)^{3} + \left(x - 8\right)^{3}

l

\left(x - 3\right)^{3} - \left(x + 2\right)^{3}

m

a^{3} + 5 a^{2} + a + 5

n

17 x^{3} + 5 x^{2} + 17 x + 5

2

Simplify the following rational expressions:

a

\dfrac{125 x^{3} + 8}{5 x + 2}

b

\dfrac{x^{3} - 125}{4 x - 20}

c

\dfrac{x^{3} - 125}{x^{2} - 25}

d

\dfrac{20 x^{4} - 35 x^{3} + 15 x^{2} + 30 x}{5 x}

3

Consider the polynomial x^{6} - y^{6}.

a

By thinking of the polynomial as \left(x^{3}\right)^{2} - \left(y^{3}\right)^{2}, fully factorise it starting with a difference of two squares.

b

By thinking of the polynomial as \left(x^{2}\right)^{3} - \left(y^{2}\right)^{3}, show that we can use a difference of two cubes to write it in the form \left(x - y\right) \left(x + y\right) \left(x^{4} + x^{2} y^{2} + y^{4}\right).

c

By comparing the results of parts (a) and (b), write down a factorisation for

x^{4} + x^{2} y^{2} + y^{4}.

Factorisation when one linear factor is known
4

The polynomial x^{3} + 5 x^{2} + 2 x - 8 has a factor of x - 1.

a

b

Hence, factorise the polynomial completely.

5

The polynomial 25 x^{3} - 50 x^{2} - 4 x + 8 has a factor of x - 2.

a

b

Hence, factorise the polynomial completely.

6

The polynomial 4 x^{3} + 21 x^{2} + 29 x + 6 has a factor of x + 3.

a

b

Hence, factorise the polynomial completely.

Finding linear factors
7

Consider the division \dfrac{2 x^{2} + 3 x + 4}{x - 3}.

a

Without actual division, find the remainder.

b

Is x - 3 a factor of the polynomial P \left( x \right) = 2 x^{2} + 3 x + 4 ?

8

Consider the division \dfrac{x^{3} - 5 x^{2} + 3 x - 2}{x + 3}.

a

Without actual division, find the remainder.

b

Is x + 3 a factor of the polynomial P \left( x \right) = x^{3} - 5 x^{2} + 3 x - 2 ?

9

Consider the expression x^{3} - 64.

a

Factorise this expression.

b

State the number of real linear factors that x^{3} - 64 has.

10

Consider the cubic x^{3} + 4 x^{2} - 11 x - 30.

a

State whether the following could be one of the factors of the cubic:

i

x - 29

ii

x + 2

iii

5 x - 1

b

Hence, factorise the cubic.

11

Consider \left( 2 x^{3} + 8 x^{2} + x + 4\right) \div \left(x + 4\right).

a

Show that x + 4 is a factor of the polynomial P \left( x \right) = 2 x^{3} + 8 x^{2} + x + 4.

b

By completing the gaps, determine the other quadratic factor.

2 x^{3} + 8 x^{2} + x + 4 = \left(x + 4\right) \left( ⬚ x^{2} + ⬚\right)

12

Consider the division \dfrac{x^{2} - 9 x + 14}{x - 2}.

a

Without actual division, find the remainder.

b

Is x - 2 a factor of the polynomial P \left( x \right) = x^{2} - 9 x + 14 ?

c

Find the other factor of x^{2} - 9 x + 14.

13

Consider the cubic x^{3} + 9 x^{2} + 26 x + 24.

a

State one linear factor of the cubic.

b

Hence, factorise the cubic.

14

Rewrite x^{3} - 3 x^{2} - 13 x + 15 as a product of linear factors.

15

The cubic x^{3} + p x^{2} + 11 x + 20 has a factor of x - 4.

a

By observation, determine the other quadratic factor of x^{3} + p x^{2} + 11 x + 20.

b

Hence, find the value of p.

c

Factorise the cubic completely.

16

The cubic - 8 - 14 x - 7 x^{2} - x^{3} can be expressed in the form \left( - 4 - x\right) \left(x^{2} + k x + 2\right).

a

Solve for the value of k.

b

Hence, factorise the cubic completely.

17

Factorise 3 x^{3} + 17 x^{2} + 28 x + 12 completely.

Factorisation by polynomial division
18

Perform the division \dfrac{z + 10}{10}.

19

Complete the following statement:

\dfrac{⬚ x^{7} - ⬚ x^{5}}{2 x^{⬚}} = 3 x^{4} - 4 x^{2}

20

What polynomial, when divided by 2 x^{2}, produces 8 x^{6} - 5 x^{4} + 6 x^{2} as a quotient?

21

Consider the division: \dfrac{x^{3} + 1}{x + 1}

What is the best way to write the polynomial x^{3} + 1 in order to keep all of the like terms aligned during the long division process?

22

Divide x^2 + 3x + 2 by x - 3 using long division. Write your answer in the form:

x^{2} + 3 x + 2 = \left(x - 3\right)Q(x) + R(x)

23

Divide x^2 - 6x + 1 by x + 3 using long division.

a

What is the remainder of this division?

b

What is the quotient of this division?

c

Rewrite x^{2} - 6 x + 1 in terms of the divisor, the quotient and the remainder:

x^{2} - 6 x + 1= D(x) \times Q(x) + R(x)

24

Consider the division: \dfrac{x^{2} + 17 x + 70}{x + 10}.

a

Use polynomial division to determine whether x + 10 a factor of the polynomial

x^{2} + 17 x + 70.

b

Find the other factor.

25

Perform the following using long division. State the quotient and remainder.

a

\left(x^{2} - 5 x + 11\right) \div \left(x + 5\right)

b

\left( 3 x^{3} - 15 x^{2} + 2 x - 10\right) \div \left(x - 5\right)

c

\left( 3 x^{3} + 2 x^{2} + 9 x + 6\right) \div \left(x^{2} + 3\right)

26

Find the value of k in the following scenarios:

a

When 3 x^{3} - 2 x^{2} - 4 x + k is divided by x - 3, the remainder is 47.

b

When 3 x^{3} + 4 x^{2} + k x - 3 is divided by x + 1, the remainder is - 5.

27

Show that x + 2 is a factor of P \left( x \right) = x^{4} + 3 x^{3} - 7 x^{2} - 27 x - 18.

28

Consider: P \left( x \right) = x^{3} + a x^{2} + b - 9 x

a

Use the fact that P \left( x \right) is divisible by x - 4 to create an equation involving a and b, with b as the subject.

b

Use the fact that P \left( x \right) is divisible by x - 3 to create another equation involving a and b, with b as the subject.

c

By using the method of substitution or otherwise, solve for the value of a.

d

Hence, solve for the value of b.

Applications
29

Evaluate 11^{3} + 6^{3} by factorising it using the sum of two cubes method.

30

The rectangle shown has an area of 5 x^{3} + 7 x^{2} - 18 x - 8

Find a polynomial expression for its length.

31

The triangle shown below has an area of 13 n^{3} + 11 n^{2} + 29 n.

Find a polynomial expression for the perpendicular height in terms of n.

32

This parallelogram has an area of 2 x^{3} + 4 x^{2} - 5 x - 1

Find a polynomial expression for the length of its base in terms of x.