How to Factor Polynomials with Higher Exponents

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By Nicole Newman – Updated Aug 30, 2022

Factoring polynomials that contain exponents higher than two is a fundamental skill that often gets overlooked after high school. Mastering this technique not only helps you identify the greatest common factor (GCF) but also equips you to simplify complex polynomials efficiently.

Factoring Polynomials of Four or More Terms

Step 1: Identify the Greatest Common Factor

The GCF is the largest expression that divides each term without a remainder. Start by selecting the lowest exponent for each variable. For example, consider the two terms 3x³ + 6x² and 6x² – 24. The GCF is 3(x + 2):

3x³ + 6x² = 3x²(x + 2)
6x² – 24 = 6(x² – 4) = 2·3(x + 2)(x – 2)

Pulling out the common factor yields 3(x + 2).

Step 2: Group the Terms

If the expression has at least four terms, group them in pairs. For x³ + 7x² + 2x + 14, create the groups (x³ + 7x²) and (2x + 14).

Step 3: Factor Within Each Group

Extract the GCF from each binomial. Using the previous example:

First group: x²(x + 7)
Second group: 2(x + 7)

Step 4: Factor Out the Common Binomial

Both groups share (x + 7). Factor it out to get (x + 7)(x² + 2).

Factoring Polynomials of Three Terms

Step 1: Extract a Common Monomial

Factor out the greatest common monomial before tackling the remaining terms. For 6x⁵ + 5x⁴ + x⁶, factor x⁴ to obtain x⁴(x² + 6x + 5).

Step 2: Factor the Trinomial Inside

When the leading coefficient is 1, look for two numbers that multiply to the constant term and add to the middle coefficient. If the leading coefficient differs from 1, find numbers that multiply to the product of the leading coefficient and constant term and sum to the middle coefficient.

Step 3: Write the Final Factorization

Place the two numbers from Step 2 into separate parentheses, ensuring the signs match the constant term. For the example, the result is x⁴(x + 5)(x + 1). Always verify by expanding the product back to the original polynomial.