7.5 General Strategy for Factoring Polynomials

General Strategy for Factoring Polynomials

Factoring polynomials means breaking down a complex expression into a product of simpler ones. It's one of the most important skills in algebra because it's how you'll solve quadratic equations, simplify rational expressions, and work with higher-level math. The challenge is that there are several different factoring methods, and knowing which one to use is half the battle. This section gives you a reliable, step-by-step strategy so you're never staring at a polynomial wondering where to start.

Methods of Polynomial Factoring

Before diving into the strategy, you need to know the tools in your toolkit.

Greatest Common Factor (GCF): Find the largest factor shared by every term and factor it out. For example, $6x^3 + 9x^2 = 3x^2(2x + 3)$. Always try this first.

Difference of Squares: When you have two perfect squares separated by a minus sign, it factors neatly:

$a^2 - b^2 = (a+b)(a-b)$

• Example: $x^2 - 9 = (x+3)(x-3)$

Note: a sum of squares ($a^2 + b^2$) does not factor over the real numbers. This is a common mistake.

Sum or Difference of Cubes: These follow specific patterns:

$a^3 + b^3 = (a + b)(a^2 - ab + b^2)$

$a^3 - b^3 = (a - b)(a^2 + ab + b^2)$

• Example: $x^3 - 8 = (x-2)(x^2+2x+4)$

A helpful mnemonic: SOAP (Same, Opposite, Always Positive). The first binomial has the same sign as the original, the second factor starts with the opposite sign, and the last term is always positive.

Perfect Square Trinomial: When the first and last terms are perfect squares and the middle term is twice their product:

$a^2 + 2ab + b^2 = (a+b)^2$

$a^2 - 2ab + b^2 = (a-b)^2$

• Example: $x^2 + 6x + 9 = (x+3)^2$ because $2 \cdot x \cdot 3 = 6x$

Factoring Trinomials (Trial and Error / AC Method): For trinomials like $ax^2 + bx + c$ that aren't perfect squares, you look for two numbers that multiply to $a \cdot c$ and add to $b$, then split the middle term and factor by grouping.

Factoring by Grouping: When you have four terms, group them in pairs, factor out the GCF from each pair, then factor out the common binomial.

• Example: $x^3 + 2x^2 + 3x + 6 = x^2(x+2) + 3(x+2) = (x+2)(x^2+3)$

Steps in the General Factoring Strategy

This is the core of the section. Follow these steps in order, every time.

1. Factor out the GCF. Look at all the terms and pull out the largest factor they share (both coefficients and variables with the lowest exponent). If the leading coefficient is negative, factor out $-1$ as part of the GCF.

   • Example: $-2x^3 + 8x^2 - 6x = -2x(x^2 - 4x + 3)$

2. Count the terms in what remains inside the parentheses.

   • Two terms: Check for difference of squares, sum/difference of cubes.
   • Three terms: Check for a perfect square trinomial. If it isn't one, use trial and error or the AC method.
   • Four or more terms: Try factoring by grouping.

3. Check each factor again. Can any of the resulting factors be factored further? If so, keep going.

   • Example: $x^4 - 81 = (x^2+9)(x^2-9) = (x^2+9)(x+3)(x-3)$
   • Here, $(x^2 - 9)$ was itself a difference of squares, so it needed one more round of factoring.

4. Verify your answer by multiplying the factors back together. You should get the original polynomial. This is the fastest way to catch mistakes.

A polynomial is fully factored when none of its factors (other than monomials) can be broken down any further.

Special Cases of Factoring

These patterns come up constantly, so recognizing them quickly will save you time.

Difference of Squares: $a^2 - b^2 = (a+b)(a-b)$

• $x^2 - 9 = (x+3)(x-3)$
• $4x^2 - 25y^2 = (2x+5y)(2x-5y)$

Remember that both terms must be perfect squares and they must be subtracted. If you see $x^2 + 16$, stop. It doesn't factor.

Perfect Square Trinomials: $a^2 \pm 2ab + b^2 = (a \pm b)^2$

To confirm you have one, check: is the middle term equal to $2 \times \sqrt{\text{first term}} \times \sqrt{\text{last term}}$?

• $x^2 + 6x + 9 = (x+3)^2$ because $2(x)(3) = 6x$ ✓
• $x^2 - 10x + 25 = (x-5)^2$ because $2(x)(5) = 10x$ ✓

Advanced Factoring Techniques

You may encounter these in later sections or more challenging problems.

Factoring by Substitution: When a polynomial has a repeated expression, replace it with a single variable to simplify. For example, to factor $x^4 - 5x^2 + 4$, let $u = x^2$. The expression becomes $u^2 - 5u + 4 = (u-1)(u-4)$. Then substitute back: $(x^2 - 1)(x^2 - 4)$. Both of those are differences of squares, so the final answer is $(x+1)(x-1)(x+2)(x-2)$.

Recognizing Prime Polynomials: Not every polynomial can be factored. After running through the strategy, if no method works, the polynomial is prime (irreducible). For instance, $x^2 + x + 1$ has no real factors. Don't waste time forcing a factorization that doesn't exist.