Expanding with the Distributive Law

Expanding means removing brackets. The distributive law says the multiplier outside visits EVERY term inside: a(b + c) = ab + ac. The picture makes it obvious: a rectangle 3 tall and (x + 4) wide has area 3(x + 4) as a whole, or 3x + 12 as two pieces — same rectangle, so 3(x + 4) = 3x + 12. Check with x = 5: 3 × 9 = 27, and 15 + 12 = 27 ✓. The bracket is a package deal — everyone inside gets multiplied.

The multiplier outside can be a term with letters. x(x + 5): the x visits both residents — x × x = x² and x × 5 = 5x — giving x² + 5x. Each visit is exactly the term multiplication from last lesson: sign, coefficients, letters. Bigger example: 2a(3a + 4b) = 6a² + 8ab. Draw the arrows from the outside term to each inside term until it's automatic — two residents, two arrows, two products.

The danger zone: a negative multiplier. The WHOLE term −3 visits each resident of (x − 4): −3 × x = −3x, and −3 × (−4) = +12 (negative × negative!). So −3(x − 4) = −3x + 12 — both signs in the answer differ from a careless guess. Special case: a bare minus is −1, so −(a − b + 3) = −a + b − 3 — every sign inside flips. Always verify with a number: at x = 10, −3(6) = −18 and −30 + 12 = −18 ✓.

Real questions chain the skills: expand every bracket, THEN collect like terms. 3(2x + 1) + 4(x − 2) = 6x + 3 + 4x − 8 = 10x − 5. The trap lives in subtraction between brackets: in 5(x + 3) − 2(x − 1), the multiplier for the second bracket is −2, so it gives −2x + 2 (not −2x − 2). Full run: 5x + 15 − 2x + 2 = 3x + 17. Expand first, breathe, then collect — never try to do both jobs in one pass.

You've used the distributive law for years without the name: 7 × 103 = 7(100 + 3) = 700 + 21 = 721 in your head, and 6 × 99 = 6(100 − 1) = 594. But its real power is PROOF. The trick 'think of a number, double it, add 6, halve it, subtract your number' always lands on 3 — because (2n + 6)/2 − n = n + 3 − n = 3, for every possible n. Trying examples suggests a trick works; expanding proves it must.

The expanding toolkit: the outside term visits every inside term; letters stack into powers on the way (x × x = x²); a negative multiplier flips the inside signs (minus × minus = plus); and multi-bracket questions are expand-first-collect-second. The safety net never changes — substitute a number into the original and your answer: if they disagree, hunt the sign error. Next lesson runs this machine in reverse: factorising. Quiz and exercises!