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Expanding Brackets

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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 13 marks

Expand and simplify

(2x+5)(x-4)(3x-1)

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

Hint

Multiply two of the brackets together first and simplify, then multiply your answer by the third bracket.

Worked solution
  1. Expand the first two brackets: (2x+5)(x-4)=2x^2-8x+5x-20=2x^2-3x-20
  2. Now multiply by (3x-1): (2x^2-3x-20)(3x-1)
  3. 2x^2\times 3x=6x^3, \ 2x^2\times(-1)=-2x^2
  4. -3x\times 3x=-9x^2, \ -3x\times(-1)=+3x
  5. -20\times 3x=-60x, \ -20\times(-1)=+20
  6. Collect like terms: 6x^3-11x^2-57x+20
  7. Answer: 6x^3-11x^2-57x+20

Question 22 marks

The expression

(x^2-4x+7)(3x+2)

is expanded and simplified.

Work out the coefficient of x^2

Hint

You don't need the whole expansion: find every pair of terms, one from each bracket, that multiplies to give an x^2 term.

Worked solution
  1. Look for the products that give x^2:
  2. x^2\times 2=2x^2
  3. -4x\times 3x=-12x^2
  4. No other pair gives x^2 (7\times 3x gives an x term, x^2\times 3x gives x^3).
  5. Add them: 2x^2-12x^2=-10x^2
  6. The coefficient is the number in front of x^2: -10

Question 3Challenge5 marks

k, a and b are constants.

(2x+k)(x-3)^2-(x+2)^3\equiv x^3+ax^2+bx-26

(a)

Work out the value of k.

2 marks

(b)

Work out the value of a.

2 marks

(c)

Work out the value of b.

1 mark

Hint

Start with the constant terms: they only involve k. Then put your value of k back in and expand everything, remembering (x-3)^2=(x-3)(x-3).

Worked solution

Part (a)

  1. Compare the constant terms (the terms with no x).
  2. (x-3)^2=x^2-6x+9, so the constant in (2x+k)(x-3)^2 is k\times 9=9k
  3. (x+2)^3 has constant term 2^3=8
  4. So 9k-8=-26
  5. 9k=-18
  6. k=-2

Part (b)

  1. Use k=-2: (2x-2)(x^2-6x+9)=2x^3-12x^2+18x-2x^2+12x-18
  2. =2x^3-14x^2+30x-18
  3. (x+2)^3=(x+2)(x^2+4x+4)=x^3+6x^2+12x+8
  4. Subtract, using brackets: 2x^3-14x^2+30x-18-(x^3+6x^2+12x+8)
  5. =x^3-20x^2+18x-26
  6. Compare the x^2 terms: a=-20

Part (c)

  1. From part (b): x^3-20x^2+18x-26
  2. Compare the x terms: b=18

More on this topic: Expanding Brackets worksheet with full solutions

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