Factoring Reveals Polynomial Structure

Factoring rewrites a sum as a product. The value stays unchanged, but the product form exposes common structure, zeros, and cancellation opportunities.

Learning objectives

You will extract greatest common factors, recognize standard identities, factor simple quadratics, and use the zero-product property.

Start with the greatest common factor

For $12x^3-18x^2$, every term contains $6x^2$:

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

Always check for a common factor first; later methods are easier after it is removed.

Identities explain patterns

Distribution verifies

$a^2-b^2=(a-b)(a+b),$ $a^2+2ab+b^2=(a+b)^2.$

Pattern recognition should be followed by multiplication back to the original expression.

Factors solve equations

If $(x+2)(x+3)=0$, the zero-product property says at least one factor is zero. Thus $x=-2$ or $x=-3$.

Check your understanding

Factor completely and solve $2x^3-8x=0$.

Show the reasoning

$2x^3-8x=2x(x^2-4)=2x(x-2)(x+2)$. Therefore $x=0,2,-2$.