Good factoring starts with looking, not computing. In Factoring Completely you learned the patterns: common factors, trinomials, differences of squares. Here you learn to spot those patterns when they are wearing a disguise, by treating a whole chunk of an expression as one quantity.
Key idea
If part of an expression repeats or is a perfect square, give it a single name, such as $u = x^2$ or $u = x + 1$. Factor the simpler expression in $u$, then put the chunk back. $x^4 - 9$ is $u^2 - 9$ with $u = x^2$, so it equals $(x^2 - 3)(x^2 + 3)$.
Why it works
Algebra rules don't care what a letter stands for. The identity $u^2 - b^2 = (u - b)(u + b)$ is true when $u$ is a number, so it is true when $u$ is any expression that stands for a number, including $x^2$ or $(x + 1)$. Renaming a chunk just lets your eyes see a familiar shape.
| Expression | Chunk | Familiar shape | Factored |
|---|---|---|---|
| $x^4 - 81$ | $u = x^2$ | $u^2 - 81$ | $(x^2 - 9)(x^2 + 9)$ $= (x - 3)(x + 3)(x^2 + 9)$ |
| $(x + 2)^2 - 25$ | $u = x + 2$ | $u^2 - 25$ | $(x - 3)(x + 7)$ |
| $(x - 1)^2 + 5(x - 1) + 6$ | $u = x - 1$ | $u^2 + 5u + 6$ | $(x + 1)(x + 2)$ |
| $53^2 - 47^2$ | numbers | $a^2 - b^2$ | $6 \cdot 100 = 600$ |
Worked examples
Worked example: A fourth power
Factor $x^4 - 16$ completely.
- Let $u = x^2$: $u^2 - 16 = (u - 4)(u + 4)$.
- Put $x^2$ back: $(x^2 - 4)(x^2 + 4)$.
- $x^2 - 4$ is again a difference of squares: $(x - 2)(x + 2)$. The sum $x^2 + 4$ does not factor over the real numbers.
Answer: $(x - 2)(x + 2)(x^2 + 4)$
Worked example: A shifted square
Factor $(x + 2)^2 - 25$.
- Let $u = x + 2$: $u^2 - 25 = (u - 5)(u + 5)$.
- Put it back: $(x + 2 - 5)(x + 2 + 5)$ $= (x - 3)(x + 7)$.
- Check by expanding both: $(x + 2)^2 - 25 = x^2 + 4x - 21$ and $(x - 3)(x + 7) = x^2 + 4x - 21$.
Answer: $(x - 3)(x + 7)$
Worked example: Arithmetic by structure
Find $53^2 - 47^2$ without a calculator.
- Difference of squares: $(53 - 47)(53 + 47) = 6 \cdot 100$.
Answer: $600$
Common mistake
Expanding everything first. Turning $(x + 2)^2 - 25$ into $x^2 + 4x - 21$ works, but it hides the pattern and gives you a harder trinomial to factor. With $x^4 - 9$, expanding doesn't even help. Look for squares (and other repeated chunks) before you multiply anything out.