Why does one technique matter this much?
A factored expression tells you where it equals zero, and almost nothing else does. That single fact is what makes it load-bearing.
If a product is zero, one of the factors must be zero — so (x − 3)(x + 2) = 0 immediately gives x = 3 or x = −2. There is no equivalent shortcut for x² − x − 6 = 0 in that form. Everything that follows uses this: solving quadratics, finding where a graph crosses the axis, working out where a rational expression is undefined, simplifying by cancelling.
It also compounds. A student who has to stop and think about x² − x − 6 does not just lose that minute; they lose it again in every Math 20-1 question that has factoring as its first step, which is most of them. Speed here buys time everywhere later.
Where it turns up
- Solving anything quadratic — projectile heights, maximum areas, break-even prices
- Graphing — factored form hands you the x-intercepts, and the vertex sits halfway between them
- Rational expressions — you can only cancel factors, so nothing simplifies until it is factored
- Math 20-1 and 30-1 — both assume this is automatic and neither reteaches it
- 1. The words, first
- 2. The order of attack
- 3. Completely means completely
- 4. What it unlocks
- 5. How to make it automatic
The words, first
The idea: A factor multiplies; a term is added. Every rule about what may be cancelled follows from that one distinction.
| Word | What it means |
|---|---|
| Term | A part separated by + or −. In 3x² + 5x − 2 there are three. |
| Factor | A part that is multiplied. In 3x(x + 2) the factors are 3, x and (x + 2). |
| Greatest common factor | The largest factor every term shares. Always taken out first. |
| Difference of squares | a² − b² = (a − b)(a + b). A sum of squares does not factor over the reals. |
| Perfect square trinomial | a² + 2ab + b² = (a + b)². The middle term is twice the product, which is the check. |
| Zero product property | If a product is zero, at least one factor is zero. The reason factoring solves equations. |
| Completely factored | No factor can be broken down further. (2x + 4)(x + 3) is not, because 2x + 4 still has a common factor. |
The order of attack
The idea: Four steps, always in this order. Skipping the first makes every later step harder.
- Greatest common factor. 3x² + 6x = 3x(x + 2). Do this even when it looks unnecessary — it makes the numbers smaller and often reveals the pattern underneath.
- Count the terms. Two terms → look for a difference of squares. Three terms → trinomial. Four terms → grouping.
- Match a pattern. x² − 25 = (x − 5)(x + 5). And 4x² − 49 = (2x − 7)(2x + 7), because the first term is a square even with a coefficient.
- Trinomial method. With a = 1, two numbers multiplying to c and adding to b. With a ≠ 1, try and check, or decompose.
Signs, as a rule rather than a guess. A positive constant means both brackets carry the sign of the middle term. A negative constant means the signs differ, and the larger number takes the sign of the middle term.
Always expand to check. Ten seconds, and it catches every sign slip. (3x + 1)(x + 2) = 3x² + 7x + 2 ✓, while (3x + 2)(x + 1) gives 5x in the middle — so which bracket gets the 2 matters.
Completely means completely
The idea: An answer that still has a common factor inside a bracket is not finished, and it usually costs a mark even though the expansion is correct.
2x² + 10x + 12. Taking out the 2 first leaves x² + 5x + 6, which factors to (x + 2)(x + 3). So:
2x² + 10x + 12 = 2(x + 2)(x + 3)
Writing (2x + 4)(x + 3) expands correctly and is not completely factored, because 2x + 4 = 2(x + 2).
A check worth doing. Look at each bracket and ask whether its terms share anything. If they do, you are not finished.
What it unlocks
The idea: Four things, and each one is a whole topic in a later course.
Solving. x² − 5x + 6 = 0 → (x − 2)(x − 3) = 0 → x = 2 or 3. The roots come out with the opposite signs to the numbers in the brackets.
Graphing. From y = (x − 2)(x − 3), the x-intercepts are 2 and 3 and the axis of symmetry is halfway between them at x = 2.5. One substitution gives the vertex, and the whole parabola is determined.
Simplifying. (x² − 9)/(x + 3) = (x − 3)(x + 3)/(x + 3) = x − 3, with x ≠ −3. Nothing cancels until both parts are factored — and only factors cancel, never terms.
Restrictions. A rational expression is undefined where its factored denominator is zero. Finding those values is finding the factors.
How to make it automatic
The idea: This is one of the few parts of mathematics where drilling genuinely is the fastest route.
Aim for recognition, not calculation. x² + 7x + 12 should arrive as (x + 3)(x + 4) without a search. That comes from having done a few hundred, not from understanding it better.
Mix the types. Practising twenty difference-of-squares in a row teaches you to recognise a page heading, not a pattern. Mixed sets are harder and are the point.
Always expand to check. It makes practice self-marking, which means you can do it without anyone else in the room.
Ten minutes a day for two weeks is enough to get this to automatic, and it stays there. It is the best return on time available in Math 10C.