Lesson 23: Factoring Polynomials

New Concept

To completely factor a polynomial of two or more terms means to express it as a product of prime polynomial factors.

Why it matters

Factoring is how we deconstruct complex expressions to reveal their simpler, fundamental components. Mastering this allows you to solve sophisticated equations, turning seemingly impossible problems into manageable steps.

What’s next

Next, you’ll learn specific patterns to factor different types of polynomials, starting with the greatest common factor.

Property: The first step in factoring any polynomial is to pull out the greatest common monomial factor (GCF). When the leading coefficient is negative, factor out the negative as well. This simplifies the remaining polynomial, making it easier to see if it can be factored further using other patterns.

Factor $4x^2 + 8x - 12$: $4(x^2 + 2x - 3)$
Factor $-6y^3 - 15y$: $-3y(2y^2 + 5)$
Factor $10a^3b^2 + 5a^2b - 15ab$: $5ab(2a^2b + a - 3)$

Think of the GCF as the team captain! It's the biggest number and variable combo that's a part of every term. By pulling it out to the front, you're simplifying the team roster, making the next play (factoring) much easier to strategize. It's the essential first move to clean up the expression and see what you're really working with.

Property: $a^2 + 2ab + b^2 = (a + b)^2$ and $a^2 - 2ab + b^2 = (a - b)^2$. A trinomial is a perfect square if its first and last terms are perfect squares, and the middle term is exactly twice the product of their square roots. Recognizing this pattern is a major shortcut.

Factor $x^2 - 14x + 49$: $(x - 7)^2$
Factor $9m^2 + 24mn + 16n^2$: $(3m + 4n)^2$
Factor $y^2 + 20y + 100$: $(y + 10)^2$

Spotting a perfect square trinomial is like finding a secret passage in a video game. If you see two perfect squares at the ends ($a^2$ and $b^2$) and the middle term is just $2ab$ (or $-2ab$), you can skip all the hard work and jump straight to the answer: $(a+b)^2$ or $(a-b)^2$.

Property

$a^2 - b^2 = (a + b)(a - b)$. This pattern applies whenever you subtract one perfect square from another. This simple but powerful rule allows you to quickly factor expressions that have only two terms separated by a minus sign, where both terms are perfect squares. It is a fundamental pattern in algebra.

Examples

Factor $x^2 - 81$: $(x + 9)(x - 9)$
Factor $16x^2 - 900$: $(4x + 30)(4x - 30)$
Factor $25y^2 - 4z^2$: $(5y + 2z)(5y - 2z)$

Explanation

This is the 'opposites attract' of algebra! When a perfect square is fighting a another perfect square ($a^2 - b^2$), they break up into two nearly identical binomials but with opposite signs: $(a+b)$ and $(a-b)$. When you multiply them back, the middle terms cancel out perfectly, leaving you right where you started. It's clean and simple!