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Multiplying & Dividing Algebraic Terms

Maths · Y8 Algebra
Year 8
Lesson Guide
Explanation
Multiplying is friendlier than adding: you can multiply ANY two terms, like or not. The method: expose the hidden × signs, then reorder — numbers with numbers, letters with letters. 3a × 4b = 3 × a × 4 × b = (3 × 4) × (a × b) = 12ab. Multiplication doesn't care about order, so you're free to shuffle. Convention: write the coefficient first and the letters alphabetically — 12ab, not ab12 or 12ba.
Your Turn
Simplify 5m × 3n, and 2 × 7k. Check: 15mn and 14k.
Watch Out
Don't confuse the rules: 3a + 4b won't combine, but 3a × 4b = 12ab. Adding needs like terms; multiplying doesn't.
Explanation
When a letter meets itself in multiplication, it stacks into a power: x × x = x2, so 4x × 3x = 12x2. Keep this razor-sharp: x + x = 2x (two copies ADDED) but x × x = x2 (two copies MULTIPLIED). At x = 3 that's 6 versus 9 — genuinely different. With more letters, count each species: 2a × 5ab = 10 × a × a × b = 10a2b. Every repeated letter climbs one power per copy.
Your Turn
Simplify 6y × 2y, and 3p × 4pq. Check: 12y2 and 12p2q.
Watch Out
4x × 3x = 12x2, NOT 12x. Both x's must appear in the answer — as a square.
Explanation
Chapter 1's sign rules ride along unchanged: same signs → positive, different signs → negative. So −3a × 4b = −12ab, and −3a × (−4b) = +12ab. The professional habit is to do the three jobs in order: sign first, then coefficients, then letters. For (−2x) × (−3x): signs — negative × negative = positive; numbers — 2 × 3 = 6; letters — x × x = x2. Answer: +6x2. Three small decisions instead of one tangled one.
Your Turn
Simplify (−5m) × 4n, and (−2y) × (−7y). Check: −20mn and 14y2.
Watch Out
THREE negatives multiply to a negative: (−2) × (−3) × (−a) = −6a. Count the minus signs — odd count means negative.
Explanation
Division becomes easy the moment you write it as a fraction. 12ab ÷ 3a = (12 × a × b)/(3 × a). Now cancel: 12 over 3 is 4, and a over a is 1 — the a's annihilate. Answer: 4b. This is exactly the fraction-cancelling you mastered in Chapter 2, with letters joining the party. Sign rules apply here too: −18xy ÷ 6x = −3y (different signs → negative).
Your Turn
Simplify 20mn ÷ 5n, and −16pq ÷ 8p. Check: 4m and −2q.
Watch Out
a ÷ a = 1, not 0. Cancelling removes the letter from view because multiplying by 1 is invisible — nothing was subtracted.
Explanation
Powers in division: expand them into copies. 15x2 ÷ 5x = (15 × x × x)/(5 × x): one x from the top cancels the x below, one x survives. Answer: 3x. Likewise 8a3 ÷ 2a = 4a2 — each cancelled pair drops the power by one. You're quietly discovering the index law xᵃ ÷ xᵇ = xᵃ-ᵇ, which gets its own lesson soon. For now: expand, cancel, collect what's left.
Your Turn
Simplify 12y3 ÷ 4y, and 9m2 ÷ 3m2. Check: 3y2 and 3.
Watch Out
x2 ÷ x = x, not x2. Exactly one copy cancels; write out the x's if you're unsure.
Explanation
Where this all lands: a rectangle 4x long and 3x wide has area 4x × 3x = 12x2 — squared lengths make areas, which is why x2 is called 'x squared'. And division runs it backwards: area 18a2 with length 6a gives width 18a2 ÷ 6a = 3a. The complete drill for any term product or quotient: sign, coefficients, letters, powers — then verify with a substitution (x = 2: 8 × 6 = 48 = 12 × 4 ✓). Quiz and exercises!
Your Turn
A box has a square base of side 2x and height 5x. Volume? Check: 2x × 2x × 5x = 20x3.
Watch Out
Areas come out in x2, volumes in x3. If your area answer has a plain x, a length went missing somewhere.
📌 Key Points — this lesson in a nutshell
  • ANY terms can be multiplied (unlike adding): numbers × numbers, letters × letters — 3a × 4b = 12ab.
  • A letter meeting itself stacks into a power: 4x × 3x = 12x2. But x + x = 2x — adding and multiplying differ.
  • Sign rules carry over: same signs → +, different signs → −. Do sign, then numbers, then letters.
  • To divide, write a fraction and cancel: 12ab ÷ 3a = 4b (a ÷ a = 1, not 0).
  • Powers cancel one copy at a time: 15x2 ÷ 5x = 3x.
  • Verify any product or quotient by substituting a small number.
Step 1: Introduction
Step 1 of 6
3a × 4b Rewrite with the hidden × signs: 3 × a × 4 × b Reorder — numbers first: 3 × 4 × a × b = 12ab Unlike ADDING, you can multiply ANY terms — like or not. Coefficients multiply; letters line up alphabetically.
4x × 3x 4 × x × 3 × x = 12 × x × x = 12x2 x × x = x2 (two copies multiplied) x + x = 2x (two copies added) — DIFFERENT! 2a × 5ab = 10a2b — count the a's: a × a = a2. Test at x = 3: x·x = 9 = x2, but x + x = 6 = 2x. Not the same.
The sign rules ride along (+3a) × (+4b) +12ab (−3a) × (+4b) −12ab (−3a) × (−4b) +12ab (−2x) × (−3x) +6x2 Same signs → positive. Different signs → negative. Do the sign FIRST, then coefficients, then letters — three small jobs instead of one big one.
12ab ÷ 3a Write as a fraction, expose the factors: 12 × a × b 3 × a Cancel: 12 ÷ 3 = 4, and a ÷ a = 1 (the a's vanish) = 4b Division IS a fraction — the ÷ sign is just a fraction lying down.
15x2 ÷ 5x 15 × x × x 5 × x One x cancels; one x survives: 3 × x = 3x x2 ÷ x = x. Each cancelled pair drops the power by one — a preview of the index laws.
Where × and ÷ earn their keep 4x 3x A = 12x2 Area = length × width = 4x × 3x = 12x2 at x = 2: 8 × 6 = 48 = 12(4) ✓ Backwards too: area 18a2, length 6a → width = 18a2 ÷ 6a = 3a. The four-step drill: sign → coefficients → letters → powers. Check any answer by substituting a small number for the letter. Quiz and exercise time!
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