Year 8 Mathematics | Victorian Curriculum 2.0
Algebraic expressions (expand & factorise)
Topic 06 | Number & Algebra | Answer key

Year 8 core - answers

Fluency

Expand

    1. 3x+123x + 12
    2. 5y−105y - 10
    3. −2a−12-2a - 12
    4. −4x+12-4x + 12
    5. 14p+714p + 7
    6. x2+5xx^2 + 5x
    7. 6m2−8m6m^2 - 8m
    8. −a2+7a-a^2 + 7a
Fluency

Expand and collect

    1. 5x+95x + 9. Method: 2x+6+3x+32x + 6 + 3x + 3.
    2. 2y+22y + 2. Method: 4y−8−2y+104y - 8 - 2y + 10.
    3. 5a−25a - 2. Method: 6a−3+5−a−46a - 3 + 5 - a - 4.
    4. 7p+27p + 2. Method: 5p+2p+6−45p + 2p + 6 - 4.
    5. x2−x+3x^2 - x + 3. Method: x2+2x−3x+3x^2 + 2x - 3x + 3.
Fluency

Factorise

    1. 3(2x+3)3(2x + 3)
    2. 5(2y−3)5(2y - 3)
    3. 4(3a+4b)4(3a + 4b)
    4. 7x(x+2)7x(x + 2)
    5. 3a(3b−4)3a(3b - 4)
    6. −3(x−3)-3(x - 3) or equivalently 3(3−x)3(3 - x)
    7. 2(x2−4x+3)2(x^2 - 4x + 3)
    8. 2m(2n+3m−1)2m(2n + 3m - 1)
Fluency

Algebraic fractions

    1. 2x+32x + 3
    2. 3a−23a - 2
    3. 2x2+3x2x^2 + 3x
    4. 4a−34a - 3. Method: divide each term by 5b5b.
Reasoning

Explain and spot the mistake

    1. Wrong. The 33 must multiply every term inside the bracket: 3(x−2)=3x−63(x - 2) = 3x - 6. Jed forgot to multiply the −2-2.
    2. Wrong. The HCF of 66 and 44 is 22. Correct: 2(3x+2)2(3x + 2), not 2(3x+4)2(3x + 4) (which expands to 6x+86x + 8).
    3. Both terms on top must be divided by 22: 2x2+62=x+3\dfrac{2x}{2} + \dfrac{6}{2} = x + 3. Dividing only the 2x2x and leaving the 66 unchanged is wrong.
    4. Many answers. Example: 6x+12=6(x+2)=3(2x+4)=2(3x+6)6x + 12 = 6(x + 2) = 3(2x + 4) = 2(3x + 6). Expanding 6(x+2)6(x + 2) returns 6x+126x + 12. ✓
Problem solving

Applications

    1. Perimeter =2(x+3)+2(5)=2x+16= 2(x + 3) + 2(5) = 2x + 16. Area =5(x+3)=5x+15= 5(x + 3) = 5x + 15.
    2. Cost =4+2k=2(2+k)= 4 + 2k = 2(2 + k) (factorised).
    3. Total change =5x−42= 5x - 42. Each gets 5x−425\dfrac{5x - 42}{5}. (When this is not an integer, the “equally” is idealised.)
    4. Factorisations: 6x+12=6(x+2)6x + 12 = 6(x + 2); 4x+8=4(x+2)4x + 8 = 4(x + 2). Both rectangles share the side length (x+2)(x + 2) - i.e. they have a side in common.

Challenge - answers

Reasoning

Harder reasoning

    1. 66. Method: (x+2)(x+3)=x2+5x+6(x+2)(x+3) = x^2 + 5x + 6; subtract x(x+5)=x2+5xx(x+5) = x^2 + 5x; x2+5x+6−x2−5x=6x^2 + 5x + 6 - x^2 - 5x = 6.
    2. 4xy(x+2y)4xy(x + 2y).
    3. First area =ab= ab. Second area =2a×b2=ab= 2a \times \dfrac{b}{2} = ab. Same.
    4. x−45x - \dfrac{4}{5}. Method: top =3x−6+2x+2=5x−4= 3x - 6 + 2x + 2 = 5x - 4; divide by 55.
Year 8 Mathematics study companion | Answer key