Math 10C · Polynomials
Expanding, and the harder direction
Multiplying out is mechanical. Factoring is the reverse, it is harder, and almost everything in Math 20-1 depends on being able to do it quickly — which is why it is worth over-practising now.
- 1. The words, first
- 2. Multiplying out
- 3. Factoring, in order
- 4. Why factoring matters later
- 5. What costs marks
The words, first
The idea: A factor multiplies; a term is added. Everything about what can be cancelled follows from that one distinction.
| Word | What it means |
|---|---|
| Term | A part of an expression separated by + or −. In 3x² + 5x − 2 there are three terms. |
| Factor | A part that is multiplied. In 3x(x + 2) the factors are 3, x and (x + 2). |
| Coefficient | The number in front of a variable. |
| Degree | The highest exponent. 3x² + 5x has degree 2. |
| Binomial / trinomial | Two terms; three terms. |
| Distributive property | a(b + c) = ab + ac. The rule behind every expansion. |
| Greatest common factor | The largest factor shared by every term. Always taken out first. |
| Difference of squares | a² − b² = (a − b)(a + b). A sum of squares does not factor over the real numbers. |
| Perfect square trinomial | a² + 2ab + b² = (a + b)². The middle term is twice the product, which is the check. |
Multiplying out
The idea: Every term in the first bracket multiplies every term in the second. A grid keeps track when the brackets get big.
Two binomials. (x + 2)(x + 5) = x² + 5x + 2x + 10 = x² + 7x + 10.
Always check with a number. Put x = 1 into both forms: (3)(6) = 18 and 1 + 7 + 10 = 18 ✓. It takes five seconds and catches every sign error.
A square is not a shortcut. (x + 3)² means (x + 3)(x + 3) = x² + 6x + 9. Writing x² + 9 is the single most common algebra error at this level, and the check with x = 1 exposes it instantly: 16 versus 10.
Bigger products. For (x + 2)(x² − 3x + 1), a grid is safer than holding six products in your head:
x³ − 3x² + x + 2x² − 6x + 2 = x³ − x² − 5x + 2
Factoring, in order
The idea: Common factor first. Then look for a pattern. Then, and only then, try the trinomial method.
- Greatest common factor. 3x² + 6x = 3x(x + 2). Skipping this makes everything after it harder, and an answer that still has a common factor inside is not completely factored.
- Difference of squares. x² − 25 = (x − 5)(x + 5), and 4x² − 49 = (2x − 7)(2x + 7) — the first term is a square even with a coefficient.
- Trinomial, a = 1. Two numbers multiplying to c and adding to b. For x² + 9x + 20 that is 4 and 5.
- Trinomial, a ≠ 1. Try and check, or decompose. For 3x² + 7x + 2, (3x + 1)(x + 2) expands correctly; (3x + 2)(x + 1) gives a middle term of 5x, so which bracket gets the 2 matters.
Signs. A positive constant means both brackets share the sign of the middle term. A negative constant means the signs differ, and the larger number takes the sign of the middle term.
Factor completely. 2x² + 10x + 12 → 2(x² + 5x + 6) → 2(x + 2)(x + 3). Stopping at (2x + 4)(x + 3) expands correctly and is not finished.
Why factoring matters later
The idea: Every technique in Math 20-1 that solves or simplifies begins by factoring.
- Solving quadratics uses the zero product property, which needs factors.
- Simplifying rational expressions cancels factors — and only factors, never terms.
- Finding non-permissible values means finding where a factored denominator is zero.
- Graphing from factored form hands you the x-intercepts directly.
Which is why speed here is worth more than it looks. A student who has to think hard about x² − x − 6 will find next year's work twice as long.
What costs marks
The idea: Four, and expanding to check catches three of them.
- (x + 3)² written as x² + 9.
- Not taking out the common factor first.
- Stopping before it is completely factored.
- Cancelling terms instead of factors. In (x + 3)/(x + 5) nothing cancels at all.