AQA Level 2 Further Maths · Level 2 Certificate · Chapter 1: Number and algebra I · Lesson 3

Expanding brackets and the binomial expansion

About 80 minutesNo calculator70 marks

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Date

Do Now

The first questions are what today's lesson needs. The others bring back earlier lessons.

A1
Needed today
Expand and simplify $(3 x - 4) (2 x + 5)$ .
[2 marks]
Answer
A2
Needed today
Simplify $(- 2 y)^{3} \times (3 y)^{2}$ .
[2 marks]
Answer
A3
Last lesson
Solve $\frac{x + 3}{4} - \frac{x - 1}{6} = 2$ .
[2 marks]
Answer
A4
From chapter 1
$a : b = 3 : 5$ . Work out $(2 a + b) : (b - a)$ in its simplest form.
[2 marks]
Answer

Example, then your turn

Level 2

Your teacher works through each example with you. Then try the one beside it.

Example 1

Expand and simplify $(x + 3) (2 x - 1) (x - 4)$ .

Hint Multiply two of the brackets first and simplify. Then multiply every term of that answer by each term of the last bracket.

B1
Expand and simplify fully $(x - 3) (2 x + 5) (3 x - 1)$ .
[3 marks]
Answer

Example 2

Expand $(3 a - 2 b)^{3}$ .

Hint Row

3

of Pascal's triangle is

1, 3, 3, 1

. Put

3 a

and

- 2 b

into

(p + q)^{3} = p^{3} + 3 p^{2} q + 3 p q^{2} + q^{3}

, with brackets round each.

B2
Expand $(2 + 5 w)^{4}$ .
[3 marks]
Answer

Example 3

Work out the first three terms, in ascending powers of $x$ , in the expansion of $(3 + 2 x)^{6}$ .

Hint Ascending powers start with

x^{0}

. The first three numbers in row

6

are

1, 6, 15

B3
Work out the first three terms, in descending powers of $x$ , in the expansion of $(2 x - 1)^{7}$ .
[3 marks]
Answer

Example 4

Work out the coefficient of $x^{3}$ in the expansion of $(5 - 2 x)^{4}$ .

Hint The

x^{3}

term uses

(- 2 x)^{3}

and

5^{1}

. Its number from row

4

(

1, 4, 6, 4, 1

) is the fourth one.

B4
Work out the coefficient of $x^{3}$ in the expansion of $(1 + 3 x)^{7}$ .
[3 marks]
Answer $\dots\dots$

Example 5

Work out the term independent of $x$ in the expansion of $(x^{2} - \frac{2}{x})^{6}$ .

Hint A term is (number)

\times (x^{2})^{6 - r} \times (- \frac{2}{x})^{r}

. The power of

x

12 - 2 r - r

. Make it

0

B5
Work out the term independent of $x$ in the expansion of $(3 x - \frac{1}{x ^{2}})^{6}$ .
[3 marks]
Answer $\dots\dots$

Practice

Level 2

C1
Expand and simplify $(3 y - 1)^{3}$ . Square the bracket first.
[3 marks]
Answer
C2
Simplify $(x^{2} + 3) (x - 2) - 2 (x - 1)^{2}$ .
[3 marks]
Answer

C3
Expand and simplify
(a)
$(5 - 2 m) (3 m + 4)$
[2 marks]
Answer
(b)
$(2 p - 3 q)^{2}$
[2 marks]
Answer
Total for C3: 4 marks
C4
Multiply $(x^{3} - 3 x + 2)$ by $(x^{2} + 2 x - 1)$ . Simplify your answer.
[3 marks]
Answer
C5
Pascal's triangle starts with row $0$ , which is $1$ . Row $1$ is $1, 1$ .
(a)
Write down row $7$ .
[1 mark]
Answer
(b)
Write down the third number in row $11$ .
[1 mark]
Answer
Total for C5: 2 marks
C6
The cuboid is $(x + 4)$ cm long, $(2 x - 1)$ cm wide and $(x - 2)$ cm high. Give each answer expanded and simplified.
Not drawn accurately
(a)
Work out the volume of the cuboid.
[3 marks]
Answer $\dots\dots$ cm³
(b)
Work out the total surface area of the cuboid.
[3 marks]
Answer $\dots\dots$ cm²
Total for C6: 6 marks

Exam-style questions

AQA exam style

D1
Ella says, "The coefficient of $x^{2}$ in the expansion of $(3 x + 2)^{4}$ is $6 \times 3 \times 2 = 36$ ."
Explain Ella's mistake. Work out the correct coefficient.
[3 marks]
Answer $\dots\dots$

D2
The coefficient of $x^{2}$ in the expansion of $(2 + a x)^{6}$ is $960$ .
(a)
Work out the two possible values of $a$ .
[3 marks]
Answer $a = \dots\dots$ or $a = \dots\dots$
(b)
$a$ is positive. Work out the coefficient of $x^{3}$ in the expansion.
[2 marks]
Answer $\dots\dots$
Total for D2: 5 marks
D3
Give each expansion in ascending powers of $x$ , simplified.
(a)
Expand $(1 + 3 x)^{4}$ .
[2 marks]
Answer
(b)
Hence write down the expansion of $(1 - 3 x)^{4}$ .
[1 mark]
Answer
(c)
Hence simplify $(1 + 3 x)^{4} - (1 - 3 x)^{4}$ .
[2 marks]
Answer
Total for D3: 5 marks
D4
$(x + a) (2 x - 3) (x + b) \equiv 2 x^{3} + p x^{2} - 7 x + 6$ , where $a$ , $b$ and $p$ are integers and $a > b$ . Work out the values of $a$ , $b$ and $p$ .
[4 marks]
Answer $a = \dots\dots b = \dots\dots p = \dots\dots$
D5
$n$ is an integer. Show that $(2 n + 1)^{3} - (2 n - 1)^{3}$ is $2$ more than a multiple of $24$ .
[3 marks]
D6
How many terms are there in the simplified expansion of $(a + b)^{9}$ ? Circle your answer. $891011$
[1 mark]

Extension

Stretch

No route is given. The hints are at the end of the sheet. Use one at a time.

E1
$x + \frac{1}{x} = 3$ . Use the expansion of $(x + \frac{1}{x})^{4}$ to work out the value of $x^{4} + \frac{1}{x ^{4}}$ .
[5 marks]
Hint 1 · What to try
Expand $(x + \frac{1}{x})^{4}$ with row $4$ : $1, 4, 6, 4, 1$ . Notice that the middle three terms contain $x^{2} + \frac{1}{x ^{2}}$ .
Hint 2 · The first line
$(x + \frac{1}{x})^{4} = x^{4} + 4 x^{2} + 6 + \frac{4}{x ^{2}} + \frac{1}{x ^{4}}$ , and squaring $x + \frac{1}{x} = 3$ gives $x^{2} + 2 + \frac{1}{x ^{2}} = 9$ .
Hint 3 · The full method
So $x^{2} + \frac{1}{x ^{2}} = 7$ , and $81 = x^{4} + \frac{1}{x ^{4}} + 4 \times 7 + 6$ , which gives $x^{4} + \frac{1}{x ^{4}} = 47$ .
Answer $\dots\dots$

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can expand and simplify three brackets, or a product of two longer brackets.
I can use a row of Pascal's triangle to expand $(a + b)^{n}$ .
I can find one term or one coefficient without the whole expansion.
I can work out an unknown from a coefficient or an identity.

Answers and mark scheme

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