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Binomial Expansion

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AQA Level 2 Further Maths practice questions, each with a hint and a full worked solution.

Have a pen and paper handy, to work out your answers.

Question 11 mark

Which of these is the term in x^3 in the expansion of (1-3x)^4?

Hint

Use the row 1\;4\;6\;4\;1 of Pascal's triangle, and remember the whole term -3x is cubed.

Worked solution
  1. Row 4 of Pascal's triangle is 1\;4\;6\;4\;1
  2. The x^3 term uses the fourth number, 4, with 1^1 and (-3x)^3
  3. (-3x)^3=-27x^3 (cube the -3 as well as the x)
  4. 4\times 1\times(-27x^3)=-108x^3
  5. Answer: -108x^3

Question 23 marks

Expand and simplify fully (2+3x)^4

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

Hint

Use the row 1\;4\;6\;4\;1 of Pascal's triangle, with powers of 2 going down and powers of 3x going up.

Worked solution
  1. Row 4 of Pascal's triangle: 1\;4\;6\;4\;1
  2. 1\times2^4=16
  3. 4\times2^3\times(3x)=96x
  4. 6\times2^2\times(3x)^2=6\times4\times9x^2=216x^2
  5. 4\times2\times(3x)^3=4\times2\times27x^3=216x^3
  6. 1\times(3x)^4=81x^4
  7. Answer: 16+96x+216x^2+216x^3+81x^4

Question 3Challenge5 marks

a is a negative constant.

In the expansion of (2+ax)^4 the coefficient of x^2 is 600

(a)

Work out the value of a.

2 marks

(b)

Work out the coefficient of x^2 in the expansion of (1+3x)(2+ax)^4

3 marks

Hint

Square the whole term ax when you find the x^2 coefficient; for part (b) you need the x term of (2+ax)^4 as well as the x^2 term.

Worked solution

Part (a)

  1. The x^2 term is 6\times2^2\times(ax)^2
  2. (ax)^2=a^2x^2, so the coefficient is 24a^2
  3. 24a^2=600
  4. a^2=25, so a=5 or a=-5
  5. a is negative, so a=-5

Part (b)

  1. With a=-5: (2-5x)^4
  2. Constant term: 2^4=16 (not needed)
  3. x term: 4\times2^3\times(-5x)=-160x
  4. x^2 term: 600x^2 (from part a)
  5. In (1+3x)(\dots) the x^2 terms come from 1\times600x^2 and 3x\times(-160x)
  6. 600-480=120
  7. Answer: 120

More on this topic: Binomial Expansion worksheet with full solutions

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