A-Level Maths (Edexcel 9MA0) · Pure Year 2 · Chapter 4: Binomial expansion · Lesson 2

Expanding (a + bx)ⁿ

115 marks

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Quick questions to start.

A1
Needed today
Find the first three terms of the expansion of $(1 + x)^{- \frac{1}{2}}$ .
[2 marks]
Answer
A2
Needed today
Work out $4^{- \frac{1}{2}}$ , $8^{\frac{2}{3}}$ and $(\frac{1}{27})^{- \frac{1}{3}}$ .
[3 marks]
Answer
A3
Common slip
Write $(9 + 3 x)^{\frac{1}{2}}$ in the form $k (1 + c x)^{\frac{1}{2}}$ .
[2 marks]
Answer
A4
Needed today
Solve $\frac{2 x}{5} < 1$ .
[2 marks]
Answer

Example, then your turn

A-Level

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

Example 1

Find the first four terms of the binomial expansion of $(4 + x)^{- \frac{1}{2}}$ , in ascending powers of $x$ . State the values of $x$ for which it is valid.

Hint Take out

4^{- \frac{1}{2}}

first:

(4 + x)^{- \frac{1}{2}} = 4^{- \frac{1}{2}} (1 + \frac{x}{4})^{- \frac{1}{2}}

B1
Find the first four terms of the binomial expansion of $(9 - 2 x)^{\frac{1}{2}}$ , in ascending powers of $x$ . State the values of $x$ for which it is valid.
[5 marks]
Answer

Example 2

Find the first three terms of the binomial expansion of $(2 + 3 x)^{- 3}$ , in ascending powers of $x$ . State the values of $x$ for which it is valid.

Hint

(2 + 3 x)^{- 3} = 2^{- 3} (1 + \frac{3 x}{2})^{- 3}

. The factor is

2^{- 3} = \frac{1}{8}

, not

2

and not

8

B2
Find the first four terms of the binomial expansion of $\frac{1}{( 3 - 2 x ) ^{2}}$ , in ascending powers of $x$ . State the values of $x$ for which it is valid.
[5 marks]
Answer

Example 3

Find the expansion of $\frac{3 + x}{4 - x}$ in ascending powers of $x$ , up to and including the term in $x^{2}$ .

Hint Write it as

(3 + x) (4 - x)^{- \frac{1}{2}}

and expand

(4 - x)^{- \frac{1}{2}}

to the

x^{2}

term first.

B3
Find the expansion of $\frac{1 + 2 x}{3 + x}$ in ascending powers of $x$ , up to and including the term in $x^{2}$ .
[5 marks]
Answer

Example 4

The expansion of $(a + 3 x)^{- 2}$ , where $a$ is a positive constant, begins $\frac{1}{4} + p x + q x^{2}$ . Find $a$ , $p$ and $q$ .

Hint The constant term is

a^{- 2}

. Set

a^{- 2} = \frac{1}{4}

first, then expand.

B4
The expansion of $(a + b x)^{\frac{1}{3}}$ begins $2 + \frac{1}{4} x + \dots$ Find $a$ and $b$ , the term in $x^{2}$ , and the values of $x$ for which the expansion is valid.
[6 marks]
Answer

Example 5

Use the first three terms of the expansion of $(8 - x)^{\frac{1}{3}}$ to find an approximation to $3 7.9$ , to 5 decimal places.

Hint

7.9 = 8 - 0.1

, so use

x = 0.1

. Check it is inside the range of validity.

B5
Use the first three terms of the expansion of $(25 + x)^{\frac{1}{2}}$ to find an approximation to $24$ , to 3 decimal places.
[4 marks]
Answer

Practice

A-Level

C1
Amir expands $4 - x$ . He writes: " $4 - x = 4 (1 - \frac{x}{4})^{\frac{1}{2}} = 4 - \frac{1}{2} x - \frac{1}{32} x^{2}$ ." Find his mistake and give the correct first three terms.
[4 marks]
Answer

C2
Find the first three terms of the expansion of $\frac{1}{5 + 2 x}$ and state the values of $x$ for which it is valid. Can the expansion be used when $x = 2$ ? When $x = 3$ ? Give a reason.
[5 marks]
Answer
C3
Raj writes: " $(2 + 3 x)^{- 1} = 2^{- 1} (1 + 3 x)^{- 1} = \frac{1}{2} - \frac{3}{2} x + \frac{9}{2} x^{2}$ ." Find his mistake. Give the correct first three terms and the values of $x$ for which they are valid.
[4 marks]
Answer
C4
(a) Find the first four terms of the expansion of $(4 + x)^{\frac{3}{2}}$ . (b) Use your expansion with $x = 0.2$ to estimate $4. 2^{\frac{3}{2}}$ . Compare with the value from a calculator.
[5 marks]
Answer

Exam-style questions

Edexcel exam style

D1
$f (x) = \frac{1}{9 - 4 x}$
(a)
Find the first four terms of the binomial expansion of $f (x)$ , in ascending powers of $x$ . Give each coefficient as a simplified fraction.
[5 marks]
Answer
(b)
Hence find the expansion of $(6 - x) f (x)$ , up to and including the term in $x^{2}$ .
[3 marks]
Answer
(c)
State the values of $x$ for which the expansion is valid. Explain why it cannot be used to estimate $\frac{1}{21}$ .
[2 marks]
Total for D1: 10 marks

D2
Tom writes: " $(8 + 3 x)^{- \frac{1}{3}} = 8^{- \frac{1}{3}} (1 + \frac{3 x}{8})^{- \frac{1}{3}} = \frac{1}{2} - \frac{1}{8} x + \frac{1}{32} x^{2}$ , valid for $∣ x ∣ < 1$ ."
Tom has made two errors.
(a)
Describe both errors.
[2 marks]
(b)
Find the first three terms of the correct expansion of $(8 + 3 x)^{- \frac{1}{3}}$ .
[4 marks]
Answer
(c)
Use your expansion to estimate $\frac{1}{3 9}$ , giving your answer to 3 decimal places.
[2 marks]
Answer
Total for D2: 8 marks
D3
Given that $(a + b x)^{- \frac{3}{2}} = \frac{1}{8} + p x + \frac{15}{256} x^{2} + q x^{3} + \dots$ , where $a$ , $b$ , $p$ and $q$ are constants and $b < 0$ , find the values of $a$ , $b$ , $p$ and $q$ , and state the values of $x$ for which the expansion is valid.
[9 marks]
Answer
D4
$g (x) = \frac{1}{( 2 - 5 x ) ^{2}}$
(a)
Find the first four terms of the binomial expansion of $g (x)$ , in ascending powers of $x$ .
[5 marks]
Answer
(b)
In the expansion of $(k + x) g (x)$ , where $k$ is a constant, there is no term in $x^{2}$ . Find $k$ .
[3 marks]
Answer
(c)
State the values of $x$ for which the expansion of $g (x)$ is valid.
[2 marks]
Answer
Total for D4: 10 marks
D5
$h (x) = (16 + x)^{\frac{1}{4}}$
(a)
Find the first three terms of the binomial expansion of $h (x)$ , in ascending powers of $x$ .
[4 marks]
Answer
(b)
Use your expansion to estimate $417$ , giving your answer to 5 decimal places. Find the percentage error of this estimate, to 2 significant figures.
[3 marks]
Answer
(c)
Explain why this expansion should not be used to estimate $440$ .
[2 marks]
Total for D5: 9 marks
D6
$p (x) = \frac{1}{3 + 2 x} + \frac{1}{3 - 2 x}$
(a)
Find the first three terms of the expansion of $\frac{1}{3 + 2 x}$ , in ascending powers of $x$ .
[3 marks]
Answer
(b)
Hence show that, for small $x$ , $p (x) \approx \frac{2}{3} + \frac{8}{27} x^{2}$ .
[3 marks]
(c)
State the values of $x$ for which this is valid, and compare the approximation with the exact value of $p (0.3)$ .
[2 marks]
Total for D6: 8 marks

Extension

Stretch

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

E1
The binomial expansion of $(a + b x)^{n}$ , where $n$ is a negative integer and $a > 0$ , begins $\frac{1}{9} + \frac{2}{9} x + \frac{1}{3} x^{2} + \dots$ Find $n$ , $a$ and $b$ . Find the coefficient of $x^{3}$ and the values of $x$ for which the expansion is valid.
[9 marks]
Hint 1 · What to try
Write the expansion as $a^{n} (1 + n c x + \frac{n ( n - 1 )}{2} c^{2} x^{2} + \dots)$ with $c = \frac{b}{a}$ .
Hint 2 · The first line
Divide each coefficient by the constant term $\frac{1}{9}$ : you get $n c = 2$ and $\frac{n ( n - 1 )}{2} c^{2} = 3$ .
Hint 3 · The full method
Substitute $c = \frac{2}{n}$ into the second equation. It becomes linear in $n$ .
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can take out a factor of aⁿ and expand (a + bx)ⁿ for negative or fractional n.
I can state the values of x for which an expansion of (a + bx)ⁿ is valid.
I can expand a product or quotient with a linear factor and find unknown constants.
I can use an expansion to approximate a root and judge its accuracy.

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