Factoring

Factoring rewrites an expression as a product, revealing common structure, zeros, simplifications, and efficient algebraic methods.

Open the full lesson →

Structure → factor → verify. An expression is scanned for common structure, rewritten as factors, and checked by multiplication. The return arrow makes verification part of the method rather than an optional final step.

Factoring rewrites a sum or difference as an equivalent product. It is reverse distribution.

Why product form matters

Product form exposes repeated factors and zeros. $x^2-5x+6$ does not visibly show its roots, but

$$ x^2-5x+6=(x-2)(x-3) $$

shows that the expression is zero at $x=2$ or $x=3$.

Always check the GCF first

Find the greatest numerical and variable factor shared by every term:

$$ 12x^3-18x^2=6x^2(2x-3). $$

Factoring is incomplete if a nontrivial common factor remains inside brackets. Include a negative GCF when it makes the leading term inside positive.

A complete example

Continue until every factor is irreducible over the number system in use.

Solving with zero-product property

If a product equals zero, at least one factor is zero:

ab=0\Rightarrow a=0\text{ or }b=0.

This works only after the equation is written with zero on one side. From $2x(x-2)(x+2)=0$, solutions are $x=0,2,-2$.

Common mistakes

Skipping the GCF. Later patterns may look harder or remain incomplete.

Factoring a sum of squares as a difference. Signs will fail when expanded.

Using zero-product property before one side is zero. A product equal to another number does not split directly.

Stopping too early. Inspect every new factor again.

Quick self-check

Is there a GCF?
Which term-count or special-product pattern appears?
Is each factor fully reduced?
Does expanding reproduce every original term?