This set has twelve original questions, ordered from easier to harder, covering all five lessons in algebraic structure. Questions 1 to 4 are warm-ups on collecting and expanding, 5 to 8 cover factorising, and 9 to 12 mix skills.
Attempt each question on paper, without a calculator, and write your working as you would in an exam. Then open the answer. Mark the ones you got wrong and use the routing list at the end.
Questions
1. Simplify 5a + 3b − 2a + 4b
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The a terms: 5a − 2a = 3a. The b terms: 3b + 4b = 7b.
3a + 7b
2. Simplify 4x²y + 3xy² − x²y − 5xy²
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The x²y terms: 4 − 1 = 3, giving 3x²y. The xy² terms: 3 − 5 = −2, giving −2xy². The two kinds of term are not like terms, so they stay separate.
3x²y − 2xy²
Check with x = 2 and y = 1: the original is 16 + 6 − 4 − 10 = 8, and 3(4) − 2(2) = 8.
3. Expand and simplify −3(2x − 5) + 4
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−3 × 2x = −6x and −3 × (−5) = +15. Then add 4: −6x + 15 + 4.
−6x + 19
Check with x = 1: the original is −3(−3) + 4 = 13, and −6 + 19 = 13.
4. Expand and simplify (x + 4)(x − 7)
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Products: x², −7x, +4x, −28. The middle terms: −7x + 4x = −3x.
x² − 3x − 28
Check with x = 2: (6)(−5) = −30, and 4 − 6 − 28 = −30.
5. Factorise completely 14m²n − 21mn²
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HCF of 14 and 21 is 7. Both terms contain m and n, each with lowest power 1. The common factor is 7mn.
14m²n ÷ 7mn = 2m, and 21mn² ÷ 7mn = 3n.
7mn(2m − 3n)
6. Factorise −8y − 12, taking out a negative common factor
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The HCF of 8 and 12 is 4. Taking out −4: −8y ÷ (−4) = 2y, and −12 ÷ (−4) = +3.
−4(2y + 3)
Check: −4 × 2y = −8y and −4 × 3 = −12.
7. Factorise x² − 8x + 15
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c is positive, so both numbers share a sign, and b is negative, so both are negative. Pairs with product 15: −1 and −15 (sum −16), −3 and −5 (sum −8).
(x − 3)(x − 5)
Check: x² − 5x − 3x + 15 = x² − 8x + 15.
8. Factorise x² + 3x − 28
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c is negative, so the signs differ, and b is positive, so the larger number is positive. Pairs with product 28: 1 and 28, 2 and 14, 4 and 7. The difference of 3 comes from 4 and 7, so the pair is −4 and +7.
(x − 4)(x + 7)
Check: x² + 7x − 4x − 28 = x² + 3x − 28.
9. Expand (2x − 3)²
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(2x − 3)² means (2x − 3)(2x − 3). Products: 4x², −6x, −6x, +9.
4x² − 12x + 9
Check with x = 2: (1)² = 1, and 16 − 24 + 9 = 1. Note that 4x² − 9 would give 7, so the middle term matters.
10. Simplify (x² − 16)/(x² − 4x) and state the values x cannot take
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Top: (x − 4)(x + 4). Bottom: x(x − 4), which is zero when x = 0 or x = 4. Cancel (x − 4).
(x + 4)/x, where x ≠ 0 and x ≠ 4
Check with x = 8: the original is 48/32 = 3/2, and 12/8 = 3/2.
11. Simplify (x² + 2x − 15)/(x² − 9) and state the values x cannot take
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Top: pairs with product −15 and sum 2 give −3 and 5, so (x − 3)(x + 5). Bottom: (x − 3)(x + 3), which is zero when x = 3 or x = −3. Cancel (x − 3).
(x + 5)/(x + 3), where x ≠ 3 and x ≠ −3
Check with x = 1: the original is −12/−8 = 3/2, and 6/4 = 3/2. With x = 0: −15/−9 = 5/3, and 5/3.
12. Expand and simplify (x + 3)(x − 3) − (x − 1)², then factorise your answer
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First part: (x + 3)(x − 3) = x² − 9. Second part: (x − 1)² = x² − 2x + 1. The minus sign outside the second bracket changes every sign inside it: x² − 9 − x² + 2x − 1.
Collect: x² − x² = 0, and −9 − 1 = −10. This leaves 2x − 10.
2x − 10 = 2(x − 5)
Check with x = 4: (7)(1) − 9 = −2, and 2(4) − 10 = −2.
If you got these wrong
Match the error to the lesson, fix the lesson, then try a fresh question a few days later.
- Questions 1 and 2, like terms merged or powers changed: go to collecting terms without changing powers.
- Questions 3, 4, 9 and 12, a lost or wrong sign after expanding: go to expanding a product containing negative terms.
- Questions 5 and 6, a factor missed or a term dropped: go to factorising with a common factor.
- Questions 7 and 8, wrong pair or wrong signs: go to factorising a quadratic and test the pair with the quadratic structure explorer.
- Questions 10 and 11, cancelled the wrong thing or no exclusions: go to simplifying an algebraic fraction with stated exclusions.
Keep a short written record of each error type, for example in the mistake log and retest queue. A pattern of the same error across questions tells you more than any single mark.
If you have worked through the lessons and the same error keeps returning, that is a good moment to consider individual help. Our teachers can read your working and name the habit in online one-to-one Mathematics tuition.