Factorising with Common Factors — Solutions

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1. Factorise by highest common numerical factor

   1. 6x + 9: 3(2x + 3)
   2. 12y − 8: 4(3y − 2)
   3. 15a + 20: 5(3a + 4)
   4. 18m − 24n: 6(3m − 4n)
   5. 14p + 21q − 7: 7(2p + 3q − 1)
   6. 30x − 45y + 15: 15(2x − 3y + 1)
   7. 8a + 12b − 16c: 4(2a + 3b − 4c)
   8. 100x − 75y: 25(4x − 3y)

2. Factorise including variables

   1. x² + 5x: x(x + 5)
   2. 3y² − 9y: 3y(y − 3)
   3. 4a³ + 6a²: 2a²(2a + 3)
   4. 8m²n − 12mn²: 4mn(2m − 3n)
   5. 5p³ − 15p² + 10p: 5p(p² − 3p + 2)
   6. 6x²y + 4xy² − 2xy: 2xy(3x + 2y − 1)
   7. 9a²b² − 3a²b: 3a²b(3b − 1)
   8. 16m³n² − 24m²n³: 8m²n²(2m − 3n)

3. Fully factorise

   1. 12x³ − 18x² + 6x: 6x(2x² − 3x + 1)
   2. 20a²b − 30ab² + 10ab: 10ab(2a − 3b + 1)
   3. −3y² + 9y: 3y(−y + 3) or −3y(y − 3)
   4. −4p³ − 8p²: −4p²(p + 2)
   5. Student’s incomplete factorisation: 4(3x² + 2x) still has x as a common factor inside the bracket. Fully factorised: 4x(3x + 2)
   6. 6a(a + 3) − 4(a + 3): (a + 3) is the common factor: (a + 3)(6a − 4) = 2(a + 3)(3a − 2)

4. Grouping method

   1. ax + 3a + bx + 3b: a(x + 3) + b(x + 3) = (a + b)(x + 3)
   2. mx − 2m + nx − 2n: m(x − 2) + n(x − 2) = (m + n)(x − 2)
   3. xy + 2x + 3y + 6: x(y + 2) + 3(y + 2) = (x + 3)(y + 2)
   4. pq − 4p − 3q + 12: p(q − 4) − 3(q − 4) = (p − 3)(q − 4)
   5. 6ab + 9a − 4b − 6: 3a(2b + 3) − 2(2b + 3) = (3a − 2)(2b + 3)
   6. 2x² + 6x + 5x + 15: 2x(x + 3) + 5(x + 3) = (2x + 5)(x + 3)

5. Factorising in context

   1. Rectangle area 6x² + 9x: 3x(2x + 3). Possible dimensions: 3x by (2x + 3), or x by (6x + 9), or 3 by (2x² + 3x).
   2. Perimeter P = 14a − 21b + 7: 7(2a − 3b + 1). With a=5, b=2: original = 70−42+7 = 35; factorised = 7(10−6+1) = 7(5) = 35 ✓
   3. Profit P = R − C: P = 12n² + 8n − 4n = 12n² + 4n = 4n(3n + 1) dollars
   4. n² + n always even: n² + n = n(n + 1). For any integer n, either n or (n+1) is even, so their product is always even.

6. Factorising with a binomial common factor

   1. 6a(a + 3) − 4(a + 3) = (a + 3)(6a − 4) = 2(a + 3)(3a − 2)
   2. x(x − 2) + 5(x − 2) = (x − 2)(x + 5)
   3. 2(a+3)(3a−2) = 2(3a²+7a−6) = 6a²+14a−12. Check: 6a(a+3)−4(a+3) = 6a²+18a−4a−12 = 6a²+14a−12 ✓

7. Fully factorise

   1. −4p³ − 8p² = −4p²(p + 2)
   2. No — 10(2a²b − 3ab² + ab) still has factor ab inside. Fully factorised: 10ab(2a − 3b + 1)
   3. 6a²b² − 9a²b + 3ab = 3ab(2ab − 3a + 1)

8. Proof by factorising

   1. n² + n = n(n + 1). One of n, (n+1) is even → product is always even ✓
   2. n² − n = n(n − 1). n and n−1 are consecutive integers → one is even → product is always even ✓

9. Perimeter, profit and factorising

   1. P = (12n² + 8n) − 4n = 12n² + 4n
   2. 4n(3n + 1)
   3. n=50: 4(50)(151) = 200 × 151 = $30 200

10. Shape perimeter investigation

    1. 7(2a − 3b + 1)
    2. 14(5) − 21(2) + 7 = 70 − 42 + 7 = 35
    3. 7(10 − 6 + 1) = 7(5) = 35 ✓
    4. 7(2a − 3b + 1) = 0 when 2a = 3b − 1. E.g. a=1, b=1. A zero perimeter is physically impossible — it would make the shape degenerate.