iGCSE (2021 Edition)

# 5.02 Expansion

Worksheet
Expansions
1

The area of the figure below is \left(y + 3\right) \left(y + 7\right). We want to find another expression for this area by finding the sum of the areas of the four smaller rectangles.

a

What is the area of rectangle A?

b

What is the area of rectangle B?

c

What is the area of rectangle C?

d

What is the area of rectangle D?

e

Hence, write an equivalent expression for the area of the whole figure.

2

Complete the following statement.

\left(v - 2\right) \left(v - 3\right)=v \left(v - 3\right)-⬚ \left(⬚\right)

3

Expand and simplify:

a

- \left(x + 5\right) \left(x + 2\right)

b

9 \left( 3 x + 8 y\right)^{2}

c

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

d

\left(p + 2\right)^{2} - \left(p + 5\right)^{2}

e

\left(\dfrac{8 x - 9}{9}\right)^{2}

f

\left(\dfrac{n}{4} - \dfrac{4}{n}\right)^{2}

g

\left(x - \dfrac{2}{x}\right) \left(x + \dfrac{2}{x}\right)

h

\left(x - 6\right) \left(x^{2} - 4 x - 9\right)

i

3 \left( 6 x + \dfrac{1}{2}\right) \left(x - 2\right) - 5

j

\left( 3 x - 5\right) \left( - 5 x^{2} + 8 x + 2\right)

k

\left(x + 4\right)^{2} \left(x - 4\right)^{2}

l

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

m

\left( 5 c^{2} - 2\right) \left( 4 c^{2} + 2 - 3 c\right)

n

\left( 2 x^{5} + y^{4}\right) \left( 3 x^{5} + 5 y^{4}\right)

o

\left( 3 v + 4\right) \left( 3 v^{3} + 3 + 5 v^{2} + 4 v^{4}\right)

p

\left( 2 v^{2} - 3 v + 9\right) \left( 3 v^{2} + v + 9\right)

q

\left( 2 x + 1\right) \left( 5 x + 7\right) \left( 2 x - 1\right)

r

\left( 3 c + 2\right) \left( 2 c^{2} + 2\right)^{2}

4

Expand and simplify:

a

\left( 2 x^{3} + 5 x^{2} - 7 x + 5\right) \left(x - 1\right) - \left(x^{2} - 2 x - 1\right) \left(x - 2\right)

b

2 \left( 2 p + 3 q\right) \left(p - q\right) - \left(p + q\right) \left( 3 p + 2 q\right)

c

\left(\left( 2 t + 9\right) - 4 r\right) \left(\left( 2 t + 9\right) + 4 r\right)

d

2 x \left(x + 7\right) \left(1 - \dfrac{7}{x}\right)

e

\dfrac{\left(a + b\right)^{2} - \left(a^{2} + b^{2}\right)}{\left(x + y\right)^{2} - \left(x^{2} + y^{2}\right)}

f

\left(x - \left(1 + \sqrt{6}\right)\right) \left(x - \left(1 - \sqrt{6}\right)\right)

5

Complete the following proof that \left(a + b\right)^{2} = a^{2} + 2 a b + b^{2}.

(a + b)^2 = (⬚)(⬚)

(a + b)^2 = a(⬚) + b(⬚)

(a + b)^2 = ⬚ + ⬚ + ⬚ + ⬚

(a + b)^2 = ⬚ + ⬚ + ⬚

6
a

Evaluate each of the following:

i
\left(2 + 5\right)^{2}
ii
2^{2} + 5^{2}
b

Evaluate each of the following:

i

\left(6 + 10\right)^{2}

ii

6^{2} + 10^{2}

c

Are there any values of a and b that will make \left(a + b\right)^{2} = a^{2} + b^{2} ?

7
a

Expand and simplify \left(t + 5\right)^{2}

b

Hence or otherwise, expand \left( - t - 5\right)^{2}

c

State whether each of the following are equal to \left( - t - 5\right)^{2}

i
t^{2} - 10 t + 25
ii
- \left(t + 5\right)^{2}
iii
\left( - t \right)^{2} + 10 t + \left( - 5 \right)^{2}
iv
\left(t + 5\right)^{2}
8

By using the fact that A^{3} = A A^{2}, expand and simplify the following:

a

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

b

\left(x - 5\right)^{3}

c

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

9

Use the fact that A^{4} = A^{2} A^{2}, to expand and simplify the following:

a

\left(x + 3\right)^{4}

b

\left( 4 x + 3 y\right)^{4}

c

\left( 4 x - 3 y\right)^{4}

10

Is the following statement true or false? \left(u + 1\right) \left(u^{2} - u + 1\right) = u^{3} + 1

11

We want to expand and simplify the product \left( 4 x - 4\right) \left(x^{2} + 4 x + 4\right).

a
i

To expand \left( 4 x - 4\right) \left(x^{2} + 4 x + 4\right), what do we multiply each term of x^{2} + 4 x + 4 by first?

ii

What do we then multiply each term of x^{2} + 4 x + 4 by?

b

What final step is required to simplify the expression?

12

Without doing the multiplication completely, find the coefficient of x in the expansion of \left(x^{2} - 3 x - 5\right) \left( 5 x - 4\right)

13

Without doing the multiplication completely, find the coefficient of x^{2} in the expansion of \left(x^{2} - 2 x - 3\right) \left( 4 x^{2} - 5 x - 4\right)

14

Consider the expansion of \left( 4 x^{3} - 4 x^{2} + a x + b\right) \left( 4 x^{2} - 2 x + 9\right).

a

In the expansion, the coefficient of x^{3} is 12. Find the value of a.

b

In the expansion, the coefficient of x^{2} is 4. Find the value of b.

15

The product of a quadratic polynomial and a cubic polynomial will have what degree?

16

Without expanding, state the degree of the polynomial P \left(x\right) = \left( 5 x^{2} - 3\right) \left( 7 x^{4} + 6 x - 9\right)^{3}.

17

P \left(x\right) is a quartic polynomial. What would be the degree of (P(x))^3 ?

Applications
18

Elizabeth wants to know the difference in the cross-sectional area of the door of her tent and the tent itself. The proportions of the tent are exactly the same as the door, as seen in the picture below. The triangular door has a length and height of w and h respectively.

The top of the tent is 9\text{ cm} higher than the door, and its ends are 4\text{ cm} longer than the ends of the door.

a

Express the height of the tent in terms of h.

b

Express the length of the tent in terms of w.

c

Hence, find a simplified expression for the difference between the cross-sectional area of the tent and the tent door.

19

Find the value of 201 \times 199 by rewriting it in the form \left(x + y\right) \left(x - y\right).

20

Consider the cube shown below:

a

Find the surface area of the cube in terms of x.

b

Find the volume of the cube in terms of x.

21

A piece of wire 90\text{ cm} long is bent into the shape of a pentagon as shown:

a

Form an expression for y in terms of x.

b

Form an expression for the area of the pentagon in terms of x.

22

Write an expression in expanded form for the product of three consecutive integers if the middle integer is m.

23

Write an expression in expanded form for the product of three consecutive multiples of 4 if the middle multiple is m.

24

Consider a rectangular cardboard measuring 10\text{ cm} by 6\text{ cm}. It is to be converted into a box with no lid by cutting out square corners measuring x\text{ cm} in length and folding up the sides.

a

Find an expression for the length of the cardboard box in terms of x.

b

Find an expression for the width of the cardboard box in terms of x.

c

Form an expression in terms of x for the volume of the cardboard box. Give your answer in expanded form.