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# AS-Level Maths | Binomial Expansion
- URL: https://www.a-level-maths-tutor.com/as-level-maths-binomial-expansion/
- Published: 2026-04-28T14:52:57.000Z
- Updated: 2026-04-28T14:53:48.000Z
- Author: Jaisul Naik
- Tags: Pure Maths

## Question 1

$(a + b)^n = a^n + \\dbinom{n}{1}a^{n-1}b + \\cdots + \\dbinom{n}{r}a^{n-r}b^r + \\cdots + b^n$

Find the first four terms in the expansion in ascending powers of $x$ of

(a) $(1 + x)^{10}$

(b) $(1 + 2x)^8$

(c) $\\left(1 - \\tfrac{1}{2}x\\right)^7$

(d) $(2 + 5x)^6$

**Answers**

(a) $1 + 10x + 45x^2 + 120x^3 + \\cdots$

(b) $1 + 16x + 112x^2 + 448x^3 + \\cdots$

(c) $1 - \\tfrac{7}{2}x + \\tfrac{21}{4}x^2 - \\tfrac{35}{8}x^3 + \\cdots$

(d) $1024 + 25600x + 288000x^2 + 1920000x^3 + \\cdots$

## Question 2

Find the coefficient indicated in the following expansions.

(a) $(1 + x)^{20}$, coefficient of $x^3$

(b) $(1 - x)^{14}$, coefficient of $x^4$

(c) $(1 + 4x)^9$, coefficient of $x^2$

(d) $(1 - 3y)^{14}$, coefficient of $y^3$

**Answers**

(a) $1140$ (b) $1001$ (c) $576$ (d) $-9828$

## Question 3

Expand each of the following in ascending powers of $x$ up to and including the term in $x^2$.

(a) $(1 + x^2)(1 - 3x)^{10}$

(b) $(1 - 2x)(1 + x)^8$

(c) $(1 + x + x^2)(1 - x)^6$

(d) $(1 + 3x - x^2)(1 + 2x)^7$

**Answers**

(a) $1 - 30x + 406x^2 + \\cdots$

(b) $1 + 6x + 12x^2 + \\cdots$

(c) $1 - 5x + 10x^2 + \\cdots$

(d) $1 + 17x + 125x^2 + \\cdots$

## Question 4

Find the term independent of $y$ in each of the following expansions.

(a) $\\left(y + \\dfrac{1}{y}\\right)^8$

(b) $\\left(2y - \\dfrac{1}{2y}\\right)^{12}$

(c) $\\left(\\dfrac{1}{y} + y^2\\right)^6$

(d) $\\left(3y - \\dfrac{1}{y^2}\\right)^9$

**Answers**

(a) $70$ (b) $924$ (c) $15$ (d) $-61236$

## Question 5

(a) Expand $(1 + x)^6$ in ascending powers of $x$ up to and including the term in $x^3$, simplifying each coefficient.

(b) By substituting a suitable value of $x$ into your answer to part (a), obtain an estimate for

 (i) $1.02^6$

 (ii) $0.99^6$

giving your answers to 4 decimal places.

**Answers**

(a) $1 + 6x + 15x^2 + 20x^3 + \\cdots$

(b)(i) $1.02^6 \\approx 1.1262$

(b)(ii) $0.99^6 \\approx 0.9415$

## Question 6

(a) Expand $(1 + 2y)^8$ in ascending powers of $y$ up to and including the term in $y^3$, simplifying each coefficient.

(b) By substituting a suitable value of $y$ into your answer to part (a), obtain an estimate for

 (i) $0.98^8$

 (ii) $1.01^8$

giving your answers to 4 decimal places.

**Answers**

(a) $1 + 16y + 112y^2 + 448y^3 + \\cdots$

(b)(i) $0.98^8 \\approx 0.8508$

(b)(ii) $1.01^8 \\approx 1.0829$

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