Factoring Polynomials

Have you ever taken something apart just to see how it was built? That’s exactly what factoring is - you’re taking a polynomial and figuring out what pieces were multiplied together to create it. If you can multiply $(x + 3)(x + 2)$ to get $x^2 + 5x + 6$, then factoring asks the reverse question: “What was multiplied together to get $x^2 + 5x + 6$?”

If factoring feels like guesswork right now, don’t worry. There are reliable patterns and strategies that make it systematic. You’ve actually been factoring since you learned that $12 = 3 \times 4$ - we’re just doing the same thing with expressions that contain variables.

Core Concepts

What Is Factoring?

Factoring is the reverse of multiplying. When you multiply $(x + 2)(x + 3)$, you get $x^2 + 5x + 6$. When you factor $x^2 + 5x + 6$, you get back $(x + 2)(x + 3)$.

Think of it like this: multiplication is building something up, factoring is taking it apart. Both skills matter, and they check each other - if you’re unsure whether you factored correctly, just multiply your factors together and see if you get the original expression.

Why does factoring matter? Because it’s one of the most powerful tools for solving equations. If you can write $x^2 + 5x + 6 = 0$ as $(x + 2)(x + 3) = 0$, you immediately know that either $x + 2 = 0$ or $x + 3 = 0$, giving you $x = -2$ or $x = -3$. Factoring transforms hard problems into easier ones.

Greatest Common Factor (GCF)

Before trying any fancy factoring techniques, always look for a Greatest Common Factor first. The GCF is the largest expression that divides evenly into every term of your polynomial.

For numbers, you already know this: the GCF of 12 and 18 is 6. For polynomials, you look at both the numbers (coefficients) and the variables.

To find the GCF of polynomial terms:

1. Find the GCF of the numerical coefficients
2. For each variable, take the lowest power that appears in all terms
3. Multiply these together

For example, to find the GCF of $6x^3$, $9x^2$, and $12x$:

• GCF of 6, 9, and 12 is 3
• The variable $x$ appears in all terms; the lowest power is $x^1$
• GCF = $3x$

Factoring Out the GCF

Once you’ve found the GCF, you “factor it out” by dividing each term by the GCF and writing the result as a product.

$$6x^2 + 9x = 3x(2x + 3)$$

You can verify this by distributing: $3x \cdot 2x + 3x \cdot 3 = 6x^2 + 9x$ ✓

Always factor out the GCF first. Sometimes that’s all you need; other times, what remains can be factored further.

Factoring $x^2 + bx + c$ (Leading Coefficient of 1)

When the coefficient of $x^2$ is 1, you’re looking for two numbers that:

• Multiply to give you $c$ (the constant term)
• Add to give you $b$ (the coefficient of $x$)

For $x^2 + 7x + 12$:

• You need two numbers that multiply to 12 and add to 7
• The pairs that multiply to 12: $(1, 12), (2, 6), (3, 4)$
• Which pair adds to 7? That’s $3 + 4 = 7$
• So $x^2 + 7x + 12 = (x + 3)(x + 4)$

Sign patterns matter:

• If $c$ is positive and $b$ is positive: both numbers are positive → $(x + _)(x + _)$
• If $c$ is positive and $b$ is negative: both numbers are negative → $(x - _)(x - _)$
• If $c$ is negative: one number is positive, one is negative → $(x + _)(x - _)$

Factoring $ax^2 + bx + c$ (Leading Coefficient ≠ 1)

When the coefficient of $x^2$ isn’t 1, factoring gets trickier. The AC method works reliably:

1. Multiply $a \times c$
2. Find two numbers that multiply to $ac$ and add to $b$
3. Rewrite the middle term using these two numbers
4. Factor by grouping

For $2x^2 + 7x + 3$:

1. $a \times c = 2 \times 3 = 6$
2. Find numbers that multiply to 6 and add to 7: that’s 1 and 6
3. Rewrite: $2x^2 + 1x + 6x + 3$
4. Group: $(2x^2 + x) + (6x + 3)$
5. Factor each group: $x(2x + 1) + 3(2x + 1)$
6. Factor out the common binomial: $(2x + 1)(x + 3)$

Check by multiplying: $(2x + 1)(x + 3) = 2x^2 + 6x + x + 3 = 2x^2 + 7x + 3$ ✓

Difference of Squares

This is one of the most recognizable patterns in algebra:

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

When you see something squared minus something else squared, it factors into the sum and difference of the square roots.

Examples:

• $x^2 - 9 = x^2 - 3^2 = (x + 3)(x - 3)$
• $4x^2 - 25 = (2x)^2 - 5^2 = (2x + 5)(2x - 5)$
• $x^4 - 16 = (x^2)^2 - 4^2 = (x^2 + 4)(x^2 - 4) = (x^2 + 4)(x + 2)(x - 2)$

Important: This only works for subtraction. A sum of squares like $x^2 + 9$ cannot be factored using real numbers.

Perfect Square Trinomials

These trinomials come from squaring a binomial:

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

To recognize a perfect square trinomial:

1. The first and last terms must be perfect squares
2. The middle term must be exactly twice the product of the square roots

For $x^2 + 6x + 9$:

• $x^2$ is a perfect square (of $x$)
• $9$ is a perfect square (of $3$)
• Is the middle term $2 \cdot x \cdot 3 = 6x$? Yes!
• So $x^2 + 6x + 9 = (x + 3)^2$

For $4x^2 - 20x + 25$:

• $4x^2 = (2x)^2$ ✓
• $25 = 5^2$ ✓
• Middle term: $2 \cdot 2x \cdot 5 = 20x$ ✓ (and it’s negative)
• So $4x^2 - 20x + 25 = (2x - 5)^2$

Factoring Completely

A polynomial is factored completely when no factor can be broken down any further. This often requires multiple steps:

1. Factor out the GCF first
2. Look at what remains - is it a special pattern?
3. Keep factoring until each factor is prime (cannot be factored further)

For $3x^3 - 12x$:

1. GCF is $3x$: $3x(x^2 - 4)$
2. What remains is a difference of squares: $3x(x + 2)(x - 2)$
3. None of these can be factored further - done!

Notation and Terminology

Term	Meaning	Example
Factor	Expression that divides evenly	Factors of $x^2 - 9$ are $(x+3)$ and $(x-3)$
GCF	Greatest Common Factor	GCF of $6x^2$ and $9x$ is $3x$
Prime polynomial	Cannot be factored further	$x^2 + 1$ (over reals)
Difference of squares	$a^2 - b^2$ pattern	$x^2 - 25 = (x+5)(x-5)$

Examples

Key Rules

Factoring Strategy Checklist

When you need to factor a polynomial, work through these steps in order:

1. Factor out the GCF first - Always start here
2. Count the terms:
   • Two terms: Check for difference of squares ($a^2 - b^2$)
   • Three terms: Check if it’s a perfect square trinomial, otherwise factor using the standard method
   • Four terms: Try factoring by grouping
3. Check if each factor can be factored further
4. Verify by multiplying - Your factors should give back the original

Special Factoring Patterns

Pattern	Formula	Example
Difference of squares	$a^2 - b^2 = (a+b)(a-b)$	$x^2 - 49 = (x+7)(x-7)$
Perfect square (plus)	$a^2 + 2ab + b^2 = (a+b)^2$	$x^2 + 10x + 25 = (x+5)^2$
Perfect square (minus)	$a^2 - 2ab + b^2 = (a-b)^2$	$x^2 - 8x + 16 = (x-4)^2$

Common Mistakes to Avoid

• Forgetting to factor out the GCF first - This is the most common oversight
• Trying to factor a sum of squares - $x^2 + 9$ cannot be factored over the real numbers
• Sign errors - Pay attention to whether numbers need to be positive or negative
• Stopping too early - Make sure each factor is prime before you’re done

Real-World Applications

Finding Dimensions from Area

If the area of a rectangle is $x^2 + 5x + 6$ square units, what could the dimensions be?

Factor: $x^2 + 5x + 6 = (x + 2)(x + 3)$

The rectangle could have dimensions $(x + 2)$ by $(x + 3)$.

Simplifying Formulas

Many physics and engineering formulas can be simplified through factoring. If you have the expression $v^2 - v_0^2$, recognizing it as a difference of squares lets you write it as $(v + v_0)(v - v_0)$, which may be more useful depending on what you’re calculating.

Solving Quadratic Equations

Factoring is often the fastest way to solve quadratic equations. For $x^2 - 5x - 14 = 0$:

Factor: $(x + 2)(x - 7) = 0$

By the zero product property, either $x + 2 = 0$ or $x - 7 = 0$, giving solutions $x = -2$ or $x = 7$.

Finding Zeros of Functions

When you have a polynomial function like $f(x) = x^3 - 4x$, factoring helps you find where the function equals zero (its x-intercepts):

$f(x) = x(x^2 - 4) = x(x + 2)(x - 2)$

The zeros are $x = 0$, $x = -2$, and $x = 2$.

Self-Test Problems

Summary

• Factoring is reverse multiplication - you’re finding what expressions were multiplied together to create a polynomial
• Always factor out the GCF first - this simplifies everything else
• For $x^2 + bx + c$: Find two numbers that multiply to $c$ and add to $b$
• For $ax^2 + bx + c$: Use the AC method - multiply $a \times c$, find factors that add to $b$, then factor by grouping
• Difference of squares: $a^2 - b^2 = (a + b)(a - b)$ - one of the most useful patterns
• Perfect square trinomials: $a^2 + 2ab + b^2 = (a + b)^2$ and $a^2 - 2ab + b^2 = (a - b)^2$
• Factor completely means no factor can be broken down further
• Always check your work by multiplying your factors - you should get back the original expression

Factoring is a skill that improves with practice. The more polynomials you factor, the faster you’ll recognize patterns and the more confident you’ll become. Remember: if you’re ever unsure whether you factored correctly, just multiply to check.