Factoring Polynomials — I can factor common monomials and...

SECTION 1

Historical Context & Motivation

Long before modern algebra existed, mathematicians were already wrestling with the idea of breaking numbers and expressions into smaller, more manageable pieces. The concept of factoring — writing an expression as a product of simpler factors — stretches back thousands of years. Ancient Greek mathematicians studied the geometric relationships behind squared quantities, and scholars in the Islamic Golden Age developed systematic rules for manipulating algebraic expressions. Today, factoring is one of the most essential tools in algebra because it lets you simplify expressions, solve equations, and reveal hidden structure in polynomials.

~300 BCE

Euclid's Elements

Euclid described the geometric equivalent of factoring, including the idea that a² − b² equals the area difference of two squares, which can be rearranged into a rectangle with sides (a + b) and (a − b).

~825 CE

Al-Khwarizmi's Algebra

The Persian mathematician al-Khwarizmi wrote one of the first algebra textbooks, systematizing methods for solving equations by rearranging and simplifying — early steps toward modern factoring techniques.

1591

Viète Introduces Symbolic Algebra

François Viète began using letters for unknowns, allowing algebraic expressions like a² − b² to be written symbolically and factored using general rules rather than specific numbers.

1700s–1800s

Formalization of Polynomial Algebra

Mathematicians like Euler and Gauss developed the theory of polynomials, proving that every polynomial can be factored into simpler pieces — a result known as the Fundamental Theorem of Algebra.

The central question factoring answers is straightforward: given a polynomial expression, can you rewrite it as a product of two or more simpler expressions? Mastering this skill opens the door to solving quadratic equations, simplifying rational expressions, and understanding how polynomial functions behave on a graph. In this lesson, you will learn two foundational factoring methods — pulling out the greatest common factor (GCF) and recognizing the difference of squares pattern.

SECTION 2

Core Principles & Definitions

Before diving into techniques, it helps to lock down the key vocabulary. A polynomial is an expression made up of terms that are added or subtracted, where each term is a number times one or more variables raised to whole-number exponents (like 3x² + 5x − 7). A monomial is a polynomial with just one term, such as 6x³. Factoring a polynomial means writing it as a product of two or more expressions, essentially reversing the process of distribution (expanding).

Greatest Common Factor (GCF)

The largest expression (number and/or variable part) that divides evenly into every term of the polynomial. Always check for a GCF first before trying any other factoring method.

Difference of Squares

A binomial of the form a² − b² that factors into (a + b)(a − b). It works because the cross terms cancel when you expand the product. Note: a sum of squares (a² + b²) does not factor over the real numbers.

Factoring Is Reversing Distribution

When you distribute 3x(x + 4) you get 3x² + 12x. Factoring goes the other direction — starting from 3x² + 12x and pulling out 3x to get 3x(x + 4). Multiplying your factors should always return the original expression.

Perfect Squares

A perfect square is an expression that is the square of another expression. Numbers like 1, 4, 9, 16, 25 and variable expressions like x², 4y², 9a⁴ are perfect squares. Recognizing them is the key to spotting difference-of-squares patterns.

✦ KEY TAKEAWAY

Think of factoring like unpacking a suitcase. When you packed (distributed), you combined everything into one bag. Factoring is opening that bag and laying out the individual items so you can see what is inside. The GCF is the biggest item common to every pocket, and the difference of squares is a special zipper pattern you can recognize on sight.

SECTION 3

Visual Explanation — Area Model

One of the best ways to understand factoring is through an area model. Imagine a rectangle whose area represents a polynomial. The length and width of the rectangle are the factors. The diagram below shows how 6x² + 9x can be represented as a rectangle with width 3x and length (2x + 3), since 3x × (2x + 3) = 6x² + 9x.

The rectangle's total area equals 6x² + 9x. Its width is the GCF (3x) and its length is the remaining factor (2x + 3). Each sub-rectangle shows one term of the polynomial.

Notice how the width of the rectangle — 3x — is the GCF shared by both terms. Once you pull out that common width, the remaining lengths of each sub-rectangle (2x and 3) combine to form the other factor. This area-model thinking works for any GCF factoring problem: the GCF is always one dimension, and the leftover pieces form the other dimension.

SECTION 4

Mathematical Framework

The two factoring techniques in this lesson each have a clean algebraic formula. Memorize these patterns and practice recognizing when each one applies.

GCF FACTORING

ab + ac = a(b + c)

Here, a is the greatest common factor of every term. You divide each term by a, then write the polynomial as a times the remaining expression. This works for any number of terms: ab + ac + ad = a(b + c + d).

DIFFERENCE OF SQUARES

a² − b² = (a + b)(a − b)

Both a² and b² must be perfect squares, and they must be connected by subtraction. If you expand (a + b)(a − b), the middle terms +ab and −ab cancel, leaving a² − b².

COMBINED APPROACH

Factor out the GCF first, then check for a difference of squares inside.

Example: 2x² − 18 → first factor out 2, giving 2(x² − 9). Then x² − 9 = (x + 3)(x − 3), so the fully factored form is 2(x + 3)(x − 3). Always start with the GCF.

⚠️ Watch Out!

A sum of squares (a² + b²) cannot be factored using real numbers. Only a difference (subtraction) of squares factors. If you see x² + 25, it is already in simplest form — don't try to factor it!

SECTION 5

Recognizing Factoring Patterns

The hardest part of factoring is often deciding which technique to use. The diagram below provides a decision flowchart. Start at the top and follow the arrows based on what you see in the expression.

Follow the flowchart from top to bottom. Always start by checking for a GCF. After factoring the GCF, inspect the remaining expression for a difference-of-squares pattern. If neither method applies, the expression may require techniques from future lessons.

Examples of each factoring pattern

Expression	GCF?	Difference of Squares?	Factored Form
4x + 12	Yes → 4	No	4(x + 3)
x² − 49	No	Yes → x² − 7²	(x + 7)(x − 7)
3x² − 75	Yes → 3	Yes → x² − 25	3(x + 5)(x − 5)
10x³ + 15x²	Yes → 5x²	No	5x²(2x + 3)

SECTION 6

Worked Examples

Example 1: Factoring Out the GCF

Factor 12x³ + 18x² − 6x completely.

Step 1 — Find the GCF of the coefficientsThe coefficients are 12, 18, and 6. The greatest common factor of these numbers is 6, because 6 divides evenly into 12 (giving 2), into 18 (giving 3), and into 6 (giving 1).

GCF of coefficients = 6

Step 2 — Find the GCF of the variable partsThe variable parts are x³, x², and x. The lowest power of x present in every term is x¹, so the variable GCF is x.

GCF of variables = x

Step 3 — Combine to get the full GCFMultiply the numeric and variable GCFs together: 6 × x = 6x.

Full GCF = 6x

Step 4 — Divide each term by the GCF12x³ ÷ 6x = 2x². Then 18x² ÷ 6x = 3x. Finally, 6x ÷ 6x = 1. Write the polynomial as the GCF times the remaining expression.

12x³ + 18x² − 6x = 6x(2x² + 3x − 1)

Step 5 — Verify by distributingExpand 6x(2x² + 3x − 1): 6x × 2x² = 12x³, 6x × 3x = 18x², 6x × (−1) = −6x. This matches the original expression, so the factoring is correct.

✓ Verified

Example 2: Difference of Squares

Factor 25x² − 16 completely.

Step 1 — Check for a GCFThe terms 25x² and 16 share no common factor other than 1 (25 and 16 have no common factor, and the second term has no variable). So there is no GCF to pull out.

No GCF

Step 2 — Identify a² and b²Is 25x² a perfect square? Yes: (5x)² = 25x². Is 16 a perfect square? Yes: 4² = 16. The expression is connected by subtraction, so this fits the pattern a² − b² where a = 5x and b = 4.

a = 5x, b = 4

Step 3 — Apply the difference of squares formulaSubstitute into (a + b)(a − b): replace a with 5x and b with 4.

25x² − 16 = (5x + 4)(5x − 4)

Step 4 — Verify by expandingUse FOIL: (5x)(5x) = 25x², (5x)(−4) = −20x, (4)(5x) = 20x, (4)(−4) = −16. The middle terms −20x and +20x cancel, leaving 25x² − 16. ✓

✓ Verified

SECTION 7

Comparing GCF and Difference of Squares

Both methods are factoring techniques, but they work in different situations and serve different purposes. The table below summarizes the key differences so you can decide quickly which method to apply — or whether you need to use both together.

Feature	GCF Factoring	Difference of Squares
When to use	All terms share a common factor (number, variable, or both)	Two terms that are both perfect squares, separated by subtraction
Number of terms	Works with any number of terms (2, 3, 4, …)	Requires exactly 2 terms (a binomial)
Formula	ab + ac = a(b + c)	a² − b² = (a + b)(a − b)
Common mistake	Forgetting to include the variable part in the GCF (e.g., only factoring out 3 instead of 3x)	Trying to factor a sum of squares (a² + b²), which does not factor over the reals
Priority	Always try GCF first — it's step one of every factoring problem	Check for this pattern after extracting the GCF

✦ KEY TAKEAWAY

Think of GCF factoring as the universal first step — like washing your hands before cooking. You always do it first, no matter what recipe (factoring technique) comes next. The difference of squares is a specialized pattern you look for only after the GCF has been handled. When both apply to the same expression, use them in sequence: GCF first, then difference of squares on whatever is left.

SECTION 8

Connection to Advanced Factoring

GCF factoring and the difference of squares are the foundation of a much larger toolkit. As you progress through Algebra 1 and into Algebra 2, you will encounter more sophisticated factoring patterns. The table below previews how today's methods connect to what comes next.

This Lesson	Coming Up
GCF: pull out common monomials	Factor by grouping: split 4-term polynomials into pairs, factor GCF from each pair, then factor the common binomial
Difference of squares: a² − b² = (a + b)(a − b)	Sum and difference of cubes: a³ + b³ = (a + b)(a² − ab + b²) and a³ − b³ = (a − b)(a² + ab + b²)
Recognizing perfect square terms	Perfect square trinomials: a² + 2ab + b² = (a + b)² and a² − 2ab + b² = (a − b)²
Factoring simple binomials	Factoring general trinomials: ax² + bx + c using trial-and-error, the AC method, or the quadratic formula

The exciting part is that every advanced technique still begins with the same first step you learned today: always factor out the GCF first. Mastering these two methods now gives you a reliable starting point for any factoring problem you will ever encounter. In particular, the difference of squares pattern reappears in calculus (when simplifying limits), in physics (when factoring kinematic equations), and throughout higher mathematics.

SECTION 9

Practice Problems

PROBLEM 1 — CONCEPTUAL

Explain in your own words why a² + b² cannot be factored using the difference of squares formula, but a² − b² can. What happens to the middle terms in each case when you try to expand?

PROBLEM 2 — BASIC CALCULATION

Factor completely: 8x⁴ − 20x³ + 4x².

PROBLEM 3 — INTERMEDIATE

Factor completely: 50x² − 72.

PROBLEM 4 — APPLIED

A square garden has side length x meters. A smaller square flower bed with side length 5 meters is removed from the center. The remaining area is given by x² − 25. Factor this expression and explain what each factor means in terms of the garden's dimensions. For what value of x does the remaining area equal zero?

PROBLEM 5 — CRITICAL THINKING

Factor completely: 2x⁴ − 32. (Hint: after your first factoring step, look to see if you can factor the result even further.)

SUMMARY

Lesson Summary

Factoring is the process of rewriting a polynomial as a product of simpler expressions. The first technique you should always try is pulling out the greatest common factor (GCF) — the largest monomial that divides evenly into every term. Find the GCF of the coefficients and then take the lowest power of each variable that appears in all terms. Write the polynomial as GCF × (remaining expression).

After extracting the GCF, check whether the remaining expression is a difference of squares — a binomial of the form a² − b², which factors into (a + b)(a − b). Remember that a sum of squares (a² + b²) does not factor over the real numbers. Always verify your work by multiplying the factors back together to confirm you get the original polynomial. These two methods form the foundation for all future factoring techniques.