How to Factor Polynomials: A Practical Checklist

Factoring writes a polynomial as a product of simpler ones. Work through a checklist in order: take out common factors, look for special patterns (difference of squares, perfect squares), factor quadratics, try grouping for four terms, and use the factor theorem for higher degrees. Always check by expanding.

Step 1: Common factor first

Take out the greatest common factor of every term:

Step 2: Special patterns

Example: $x^4 - 16 = (x^2 - 4)(x^2 + 4) = (x - 2)(x + 2)(x^2 + 4)$.

$x^2 + 4$ does not factor further over the real numbers, since a sum of squares has no real roots.

Step 3: Quadratics

For $x^2 + bx + c$, find two numbers that multiply to $c$ and add to $b$:

For $ax^2 + bx + c$ with $a \ne 1$, use the ac method: find numbers multiplying to $ac$ and adding to $b$, split the middle term, then group.

Step 4: Grouping (four terms)

Step 5: Factor theorem for higher degrees

If $f(r) = 0$, then $(x - r)$ is a factor. Test divisors of the constant term, divide out, and repeat. See how to solve cubic equations.

Common mistakes

• Missing the common factor, which makes later steps harder.
• Writing $x^2 + 4 = (x + 2)^2$ (that equals $x^2 + 4x + 4$).
• Stopping too early: $(x^2 - 4)$ still factors.

Practice questions

1. $3x^2 - 12$
2. $x^2 + 7x + 12$
3. $2x^2 + 7x + 3$

Answers: 1) $3(x - 2)(x + 2)$ 2) $(x + 3)(x + 4)$ 3) $(2x + 1)(x + 3)$

The polynomial factoring calculator runs this whole checklist and checks the result by expanding it. To go the other way, use the expand calculator.

Frequently asked questions

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