Factoring polynomials — Mathematics, 14–17 years

Factoring rewrites a complicated polynomial as a product of simpler pieces. That hidden structure can make equations, graphs and calculations much easier to understand.

Products reveal structure

Factoring is the reverse of expanding brackets. Instead of turning (x + 3)(x + 2) into x² + 5x + 6, you look at x² + 5x + 6 and find the two simpler factors. The product form often exposes important values and patterns.

Why factor at all?

Expanded expressions are useful for adding and comparing, but products are useful for finding when something becomes zero. Factoring developed as a way to expose repeated building blocks, solve polynomial equations and simplify expressions without changing their value.

Factoring a quadratic

Factor x² + 7x + 12. Look for two numbers whose product is 12 and whose sum is 7: they are 3 and 4. Therefore x² + 7x + 12 = (x + 3)(x + 4). To check, multiply the brackets: x² + 4x + 3x + 12 gives the original expression.

Matching only one clue

A common mistake is choosing numbers that multiply correctly but add with the wrong sign. For x² − 5x + 6, 2 and 3 work only as (x − 2)(x − 3), because their product is positive but their sum in the middle term must be negative. Signs carry meaning here.

Finding hidden zeros

Factored polynomials help reveal where a graph crosses the horizontal axis. From (x − 2)(x + 5) = 0, one factor must be zero, so x = 2 or x = −5. The same idea appears in models of areas, motion and other changing quantities.

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