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Factorising

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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 12 marks

Factorise fully 45a^2b-20b^3

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

Hint

Take out the highest common factor of the numbers and the letters first, then look at what is left inside the bracket.

Worked solution
  1. The highest common factor of 45a^2b and 20b^3 is 5b
  2. 45a^2b-20b^3=5b(9a^2-4b^2)
  3. 9a^2-4b^2=(3a)^2-(2b)^2 is a difference of two squares
  4. 9a^2-4b^2=(3a-2b)(3a+2b)
  5. Answer: 5b(3a-2b)(3a+2b)

Question 22 marks

c and k are constants.

8x^2+kx-15\equiv(4x-3)(2x+c)

Work out the value of k.

Hint

Expand the right-hand side and compare the constant terms first to find c.

Worked solution
  1. Constant terms: -3\times c=-15, so c=5
  2. Expand (4x-3)(2x+5)=8x^2+20x-6x-15
  3. =8x^2+14x-15
  4. Compare the x terms: k=14

Question 3Challenge6 marks

(a)

Simplify fully \dfrac{12x^3-27x}{6x^2-5x-6}

3 marks

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

(b)

Solve x(2x-3)^3=5(2x-3)^2

3 marks

Give every value, separated by commas

Hint

In both parts, look for a factor common to every term before doing anything else; in (b) don't cancel it.

Worked solution

Part (a)

  1. Numerator: take out 3x: 12x^3-27x=3x(4x^2-9)
  2. 4x^2-9 is a difference of two squares: 3x(2x-3)(2x+3)
  3. Denominator: 6x^2-5x-6=(2x-3)(3x+2)
  4. Cancel the common factor (2x-3)
  5. Answer: \dfrac{3x(2x+3)}{3x+2}

Part (b)

  1. Do not divide both sides by (2x-3)^2: that loses a solution
  2. Rearrange: x(2x-3)^3-5(2x-3)^2=0
  3. Take out the common factor (2x-3)^2: (2x-3)^2\left[x(2x-3)-5\right]=0
  4. Simplify the bracket: (2x-3)^2(2x^2-3x-5)=0
  5. Factorise: (2x-3)^2(2x-5)(x+1)=0
  6. Answer: x=\frac32, x=\frac52, x=-1

More on this topic: Factorising worksheet with full solutions

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