A pattern-first guide to factoring polynomials and checking every factorization by multiplying back.
Factoring reverses multiplication
To factor an expression is to rewrite it as a product of simpler expressions. Because expansion and factoring are reverse processes, the fastest verification is to multiply the proposed factors and confirm that the original polynomial returns.
Always look for a greatest common factor first. In 6x² + 9x, both terms share 3x, so the expression factors as 3x(2x + 3). Removing the common factor first often reveals another pattern.
Difference of squares
A difference of squares contains two perfect squares separated by subtraction. The middle terms cancel when the conjugate factors are expanded. The pattern does not apply to a sum of squares over the real numbers.
For 9x² − 25, the square roots of the terms are 3x and 5. Therefore 9x² − 25 = (3x − 5)(3x + 5). Expanding gives 9x² + 15x − 15x − 25, so the middle terms cancel.
Perfect-square trinomials
A perfect-square trinomial has square first and last terms, while the middle term equals twice their square roots multiplied together. The sign of the middle term determines the sign inside the repeated binomial.
For x² + 10x + 25, the outer square roots are x and 5, and 2(x)(5) = 10x. The factorization is (x + 5)². Similarly, x² − 10x + 25 = (x − 5)².
Factor a monic quadratic trinomial
For x² + bx + c, find two numbers whose product is c and whose sum is b. In x² − 5x + 6, the numbers are −2 and −3 because their product is 6 and their sum is −5. Therefore the expression is (x − 2)(x − 3).
When the leading coefficient is not one, use factor pairs or split the middle term. For 2x² + 7x + 3, multiply a and c to get 6. The numbers 6 and 1 sum to 7, so split 7x as 6x + x, group, and factor: 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3).
A reliable factoring checklist
Put terms in descending powers, remove the greatest common factor, count the terms, and test the matching pattern. Two terms may form a difference of squares. Three terms may form a perfect square or quadratic trinomial. Four terms may allow grouping.
Do not stop until expansion reproduces every coefficient and sign. If no rational pattern works, the polynomial may be prime over the rational numbers even though it factors with irrational or complex values.
How to factor quadratic expressions completely
Complete factoring means every available common factor and rational polynomial factor has been removed. For 6x² + 9x, taking out 3x gives 3x(2x + 3). Writing only x(6x + 9) is equivalent, but it is not completely factored because the coefficients inside still share 3.
For x³ − x, first take out x to get x(x² − 1), then use the difference of squares formula on the remaining quadratic: x(x − 1)(x + 1). A factoring calculator should not silently label an unchanged factorable expression as complete; multiply its result back and check whether another GCF or special pattern remains.
Factoring practice organized by pattern
Practise one structure at a time: factor 12x² + 18x using a GCF, 16x² − 81 using conjugates, x² + 12x + 36 as a perfect square, x² − x − 12 as a monic trinomial, and 2x² + 7x + 3 as a non-monic trinomial. State why each factoring formula applies before using it.
Then mix the patterns and require an expansion check. When students ask how to factor quadratic expressions step by step, this is the decision process they need: take out the GCF, recognize the term pattern, choose factor signs, repeat if possible, and verify the complete product against the original polynomial.