Factoring a polynomial means rewriting it as a product of simpler polynomials or factors. The process depends on the type of polynomial, but there are systematic steps you can follow to factor most polynomials completely.
Steps
Look for a greatest common factor (GCF) first. Check if all terms share a common number or variable, and factor it out.
Count the number of terms. If there are two terms, check for difference of squares (a² - b²), sum or difference of cubes. If there are three terms, look for a trinomial pattern. If there are four or more terms, try grouping.
For trinomials in the form x² + bx + c, find two numbers that multiply to c and add to b. For ax² + bx + c where a is not 1, use the AC method or trial and error to find factors.
If grouping, pair terms and factor out the GCF from each pair, then factor out the common binomial.
Check if any factors can be factored further, and continue until all factors are prime (cannot be factored anymore).
Always verify your answer by multiplying the factors back together to ensure you get the original polynomial.
Worked example
Factor 2x³ - 8x. First, factor out the GCF of 2x: 2x(x² - 4). Then recognize x² - 4 as a difference of squares: 2x(x + 2)(x - 2). Check: 2x(x + 2)(x - 2) = 2x(x² - 4) = 2x³ - 8x ✓
Remember
Always start by factoring out the GCF, then use the appropriate technique based on the number of terms and patterns you recognize.