NCEA Level 1Factorising expressionsAS 91947

Factorising: 7 exam questions solved step by step

What this lesson covers

  • Factorising is expanding in reverse, and every factorisation can be checked by expanding it back.
  • Taking out the highest common factor first makes every later step simpler.
Key idea

7 questions, one order

  • 1. Common factor? Take it out first.
  • 2. Two terms: difference of squares?
  • 3. Three terms: is a equal to 1?
  • 4. Expand back to check.
Your turn

Question 1 · Achieved

  • Factorise fully 12x²y − 18xy²
Worked example

Q1: 12x²y − 18xy²

  • Numbers, then letters.
  • Lowest power of each.
  1. HCF of 12 and 18 = 6
  2. lowest powers: x and y
  3. HCF = 6xy
  4. 12x²y ÷ 6xy = 2xdivide each term
  5. 18xy² ÷ 6xy = 3y
  6. 6xy(2x − 3y)
  7. 6xy(2x − 3y) = 12x²y − 18xy²
Your turn

Question 2 · Achieved

  • Factorise x² − 3x − 28
Worked example

Q2: x² − 3x − 28

  • Read the signs first.
  • Expand back to check.
  1. multiply to −28, add to −3
  2. c < 0: opposite signs
  3. −7 × 4 = −28, −7 + 4 = −3
  4. (x − 7)(x + 4)
  5. (x − 7)(x + 4) = x² + 4x − 7x − 28
  6. = x² − 3x − 28
Your turn

Question 3 · Merit

  • Factorise fully 3x² − 48
  • Careful: 48 is not a square number.
Worked example

Q3: 3x² − 48

  • Common factor first.
  • Then two squares.
  1. 3x² − 48 = 3(x² − 16)HCF 3
  2. x² = x × x, 16 = 4²
  3. x² − 16 = (x + 4)(x − 4)difference of squares
  4. 3(x + 4)(x − 4)keep the 3
  5. (x + 4)(x − 4) = x² − 16middle terms cancel
  6. 3(x² − 16) = 3x² − 48
Your turn

Question 4 · Merit

  • Factorise 2x² + 11x + 12
Worked example

Q4: 2x² + 11x + 12

  • a = 2: use ac.
  • Split the middle term.
  1. ac = 2 × 12 = 24
  2. 8 × 3 = 24, 8 + 3 = 11
  3. 2x² + 8x + 3x + 12split the middle
  4. 2x(x + 4) + 3(x + 4)factorise in pairs
  5. (2x + 3)(x + 4)common bracket
  6. expand: 2x² + 8x + 3x + 12
  7. = 2x² + 11x + 12
Your turn

Question 5 · Merit, in context

  • A rectangular garden at a marae
  • has area (x² + 9x + 20) m².
  • Its width is (x + 4) m. Find its length,
  • and an expression for its perimeter.
Worked example

Q5: the garden plot

  • Area = length × width.
  • Answer with units.
  1. Area = (x + 4)(x + 5)4 × 5 = 20, 4 + 5 = 9
  2. length = (x + 5) mwidth is (x + 4) m
  3. P = 2(x + 4) + 2(x + 5)
  4. = 2x + 8 + 2x + 10
  5. = (4x + 18) mcollect like terms
  6. x = 2: 6 × 7 = 42 = 4 + 18 + 20
  7. x = 2: 12 + 14 = 26 = 8 + 18
  8. Length (x + 5) m, P = (4x + 18) m
Your turn

Question 6 · Excellence

  • Factorise fully 4x² − 10x − 6
  • Careful: fully.
Worked example

Q6: 4x² − 10x − 6

  • Take out the 2 first.
  • Then the ac method.
  1. 4x² − 10x − 6 = 2(2x² − 5x − 3)
  2. ac = 2 × −3 = −6
  3. −6 × 1 = −6, −6 + 1 = −5
  4. 2x² − 6x + x − 3split the middle
  5. 2x(x − 3) + 1(x − 3)factorise in pairs
  6. (2x + 1)(x − 3)common bracket
  7. 2(2x + 1)(x − 3)keep the 2
  8. (4x + 2)(x − 3)4x + 2 still has a 2
  9. 2(2x² − 5x − 3) = 4x² − 10x − 6
Your turn

Question 7 · Excellence

  • n is a whole number.
  • Use factorising to show that
  • n² + 3n + 2 is always even.
Worked example

Q7: always even

  • Examples are not a proof.
  • Read the factors.
  1. n² + 3n + 2 = (n + 1)(n + 2)1 × 2 = 2, 1 + 2 = 3
  2. n + 1, n + 2 are consecutive
  3. one of them must be even
  4. even × whole number = even
  5. so n² + 3n + 2 is always even
  6. n = 5: 25 + 15 + 2 = 42 = 6 × 7
Summary

The order that solved all seven

  • Common factor, then the right method, then expand back.
StepHow
HCF firstnumbers, then lowest powers
Two termsdifference of two squares
a is 1multiply to c, add to b
a is not 1split the middle using ac
Fullyno bracket with a factor left
Checkexpand back to the start

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