By C.D. Crowder • Updated Aug 30 2022
Polynomials consist of multiple algebraic terms. Factoring them simplifies solving and reveals their underlying structure. A fully factored polynomial is expressed as a product of simpler factors—no addition, subtraction, or division remains. By applying the techniques introduced in early math courses, factoring becomes an intuitive and enjoyable skill.
Identify the greatest common factor (GCF) shared by all terms. For example, in the polynomial 5xy + 35y + 10y², the GCF is 5y. Similarly, the expression 5(x + y) – 2x(x + y) shares the factor (x + y).
Factor out the GCF. This yields 5y(x + 7 + 2y) for the first example and (x + y)(5 – 2x) for the second.
Verify the factorization by expanding the product back to the original polynomial. A successful expansion confirms the accuracy of your factors.
When a polynomial has four terms with no obvious GCF, group them strategically.
Separate the terms into two groups: the first two and the last two. For instance, x³ + 5x² + 2x + 10 becomes (x³ + 5x²) + (2x + 10).
Find the GCF within each group. Using the example, we get x²(x + 5) + 2(x + 5).
Factor out the common binomial factor—here, (x + 5)—to obtain (x + 5)(x² + 2).
Finally, combine the remaining terms: (x² + 2)(x + 5) is the fully factored form.
Check your work by multiplying the factors to ensure you retrieve the original polynomial.
Some polynomials resist factoring via the GCF or grouping methods. In such cases, synthetic division or quadratic techniques may be required, and a complete factorization might still be impossible.
