Teach Me Maths

Factorising into Single Brackets

Question 11 mark

Factorise fully 6x+18.

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

Hint

Look for the largest number that divides both coefficients.

Worked solution

Factorise fully 6x+18.

  1. The highest common factor of 6 and 18 is 6.
  2. 6x\div6=x and 18\div6=3, so the factorisation is 6(x+3).

Question 21 mark

Factorise fully 8y-12.

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

Hint

The subtraction sign must stay with the second term.

Worked solution

Factorise fully 8y-12.

  1. The highest common factor is 4.
  2. Dividing each term by 4 gives 2y and -3. Therefore 8y-12=4(2y-3).

Question 31 mark

Factorise fully ab+ac.

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

Hint

Identify the letter appearing in every term.

Worked solution

Factorise fully ab+ac.

  1. Both terms contain the factor a.
  2. Removing one a from each term leaves b+c, so the answer is a(b+c).

Question 41 mark

Factorise fully t^2+7t.

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

Hint

A square contains two factors of the letter.

Worked solution

Factorise fully t^2+7t.

  1. Each term contains t.
  2. t^2\div t=t and 7t\div t=7, giving t(t+7).

Question 52 marks

Factorise fully 12p^2+18p.

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

Hint

Find the common numerical factor and common letter factor.

Worked solution

Factorise fully 12p^2+18p.

  1. The coefficients share 6 and both terms contain p, so take out 6p.
  2. 12p^2\div6p=2p and 18p\div6p=3.
  3. The fully factorised expression is 6p(2p+3).

Question 61 mark

Factorise fully 21+14k.

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

Hint

The constant can be the first term inside the bracket.

Worked solution

Factorise fully 21+14k.

  1. Both terms are divisible by 7.
  2. 21\div7=3 and 14k\div7=2k. The answer is 7(3+2k).

Question 72 marks

Factorise fully 15a^2-10a.

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

Hint

Include the sign when dividing each term by the common factor.

Worked solution

Factorise fully 15a^2-10a.

  1. The highest common factor is 5a.
  2. The terms remaining are 3a and -2, so the answer is 5a(3a-2).

Question 81 mark

Factorise fully 18r+30.

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

Hint

Taking out only 2 or 3 would not finish the factorisation.

Worked solution

Factorise fully 18r+30.

  1. The highest common factor of 18 and 30 is 6.
  2. 18r\div6=3r and 30\div6=5.
  3. The answer is 6(3r+5); no further common factor remains inside.

Question 92 marks

Factorise fully 10xy+15xz.

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

Hint

Only letters found in every term can be taken out.

Worked solution

Factorise fully 10xy+15xz.

  1. Both coefficients share 5 and both terms contain x.
  2. Divide by 5x: the remaining terms are 2y and 3z.
  3. Hence the answer is 5x(2y+3z).

Question 102 marks

Factorise fully 14m^3-21m^2.

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

Hint

Use the smallest power appearing in both terms.

Worked solution

Factorise fully 14m^3-21m^2.

  1. The common numerical factor is 7 and the common letter factor is m^2.
  2. 14m^3\div7m^2=2m and 21m^2\div7m^2=3.
  3. The answer is 7m^2(2m-3).

Question 112 marks

Factorise fully 12x+18y-30.

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

Hint

Check all three terms before choosing the common factor.

Worked solution

Factorise fully 12x+18y-30.

  1. The three coefficients share the factor 6.
  2. Dividing gives 2x, 3y and -5.
  3. The fully factorised expression is 6(2x+3y-5).

Question 122 marks

Factorise fully q^3+4q^2-2q.

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

Hint

Use a factor shared by every term, including the last.

Worked solution

Factorise fully q^3+4q^2-2q.

  1. Each term contains at least one q, but the last term has only one.
  2. Take out q to leave q^2+4q-2.
  3. The answer is q(q^2+4q-2).

Question 131 mark

Factorise fully -12x-18.

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

Hint

A negative common factor can make the bracket easier to read.

Worked solution

Factorise fully -12x-18.

  1. A common factor of -6 makes the terms inside positive.
  2. (-12x)\div(-6)=2x and (-18)\div(-6)=3.
  3. So -6(2x+3) is fully factorised. The equivalent form 6(-2x-3) is also correct.

Question 142 marks

Find the missing number.

16x+24=8(2x+\square) for every x.

Hint

Compare the constant terms on the two sides.

Worked solution

Find the missing number.

16x+24=8(2x+\square) for every x.

  1. The constant term on the right is 8\times\square.
  2. It must equal 24, so \square=24\div8=3.
  3. Check: 8(2x+3)=16x+24.

Question 151 mark

Nina factorises 20x^2+30x as 5x(4x+6). Which statement is correct?

Hint

Check both whether it expands correctly and whether it is fully factorised.

Worked solution

Nina factorises 20x^2+30x as 5x(4x+6). Which statement is correct?

  1. Her expression is not equivalent to the original.
  2. Her expression is equivalent, but the bracket still has a common factor of 2.
  3. She should replace x^2 with 2x.
  4. No letter can be taken out as a factor.
  1. Expanding Nina’s answer gives the original expression, so it is equivalent.
  2. However, 4x+6=2(2x+3), so the complete factorisation is 10x(2x+3).

Question 16Challenge2 marks

Factorise fully 12x^3y-18xy^2.

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

Hint

Take out only the powers shared by both terms.

Worked solution

Factorise fully 12x^3y-18xy^2.

  1. The numerical common factor is 6. Both terms contain x and y.
  2. Dividing by 6xy leaves 2x^2 and -3y.
  3. The answer is 6xy(2x^2-3y). Expanding restores both original terms.

Question 17Challenge3 marks

A rectangle has area (8x^2+20x) cm², where x>0. One side is shown in the diagram.

Find a simplified expression for the length L, in centimetres.

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

Hint

Find the common factor that matches the given side.

Worked solution

A rectangle has area (8x^2+20x) cm², where x>0. One side is shown in the diagram.

Find a simplified expression for the length L, in centimetres.

  1. Factorise the area: 8x^2+20x=4x(2x+5).
  2. Area equals the product of the side lengths. The given side is 4x, so the other side is L=2x+5.
  3. Check: multiplying 4x by 2x+5 gives the stated area.

Question 18Challenge3 marks

Factorise fully 12a^3b-18a^2b^2+6a^2b.

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

Hint

Check what remains when the last term is divided by the common factor.

Worked solution

Factorise fully 12a^3b-18a^2b^2+6a^2b.

  1. The shared factor is 6a^2b.
  2. The three quotients are 2a, -3b and 1.
  3. Therefore the answer is 6a^2b(2a-3b+1). The final term inside is 1, not 0.

Question 19Challenge3 marks

The identity Px(3x+Q)=18x^2-30x holds for every x.

(a)

Find P.

2 marks

(b)

Find Q.

1 mark

Hint

Expand the left side to compare like terms.

Worked solution

The identity Px(3x+Q)=18x^2-30x holds for every x.

Part (a)

Find P.

  1. The squared term after expansion is 3Px^2.
  2. Thus 3P=18, so P=6.

Part (b)

Find Q.

  1. The linear term is PQx. With P=6, its coefficient is 6Q.
  2. Set 6Q=-30, giving Q=-5.
  3. Check: 6x(3x-5)=18x^2-30x.

Question 20Challenge3 marks

Factorise fully (12x^2+8x)+(6x^2-2x).

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

Hint

Simplify the sum before looking for the common factor.

Worked solution

Factorise fully (12x^2+8x)+(6x^2-2x).

  1. First collect like terms: 12x^2+6x^2=18x^2 and 8x-2x=6x.
  2. The sum is 18x^2+6x. Its highest common factor is 6x.
  3. The fully factorised result is 6x(3x+1).

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