Factoring

Overview

Factoring is the process of rewriting an expression as a product of simpler expressions called factors. In Algebra I, we focus on factoring polynomials. Factoring is useful because it reveals structure, makes expressions easier to work with, and is often the key step in solving equations.

For example:
$$
x^2 - 5x + 6 = (x - 2)(x - 3)
$$
Both sides are equal, but the factored form makes the zeros of the expression ($x=2$ and $x=3$) visible.

This chapter focuses on the general idea and main strategies of factoring. Specific common patterns and special types (like common factors or quadratics) are handled in their own subchapters.

What It Means to Factor a Polynomial

To factor a polynomial means to write it as a product of polynomials of lower degree, usually with integer (or sometimes rational) coefficients, and to do this as completely as possible.

An expression is called fully factored (or factored completely) when it is written as a product of factors and no factor can be factored further (within the kind of factoring we are using).

Example:

• $6x^2 - 24x = 6x(x - 4)$ is a factorization.
• $6x(x - 4)$ is fully factored if we are working over integers, because:
• $6$ cannot be broken into more “useful” integer factors for algebraic purposes,
• $x$ is already a single variable factor,
• $x - 4$ is linear (degree $1$), and cannot be factored further with integer coefficients.

Factoring is closely related to expanding. Expanding goes from product to sum, factoring goes from sum to product:

• Expanding: $(x - 2)(x - 3) \to x^2 - 5x + 6$
• Factoring: $x^2 - 5x + 6 \to (x - 2)(x - 3)$

Why Factoring Matters

Factoring is important in many algebra topics, especially:

1. Solving polynomial equations

If you want to solve an equation like:
$$
x^2 - 5x + 6 = 0
$$
factoring gives:
$$
(x - 2)(x - 3) = 0
$$
Then you can use the zero product property:

If $AB = 0$, then $A = 0$ or $B = 0$.

So:
$$
x - 2 = 0 \quad \text{or} \quad x - 3 = 0
$$
which gives $x = 2$ or $x = 3$.

2. Understanding graphs of polynomials

When a polynomial is factored, you can read off its zeros. For example:
$$
y = (x + 1)(x - 4)
$$
This graph crosses the $x$-axis at $x = -1$ and $x = 4$.

3. Simplifying rational expressions

For expressions like:
$$
\frac{x^2 - 9}{x^2 - 3x}
$$
factoring numerator and denominator can allow simplification:
$$
\frac{(x - 3)(x + 3)}{x(x - 3)} = \frac{x + 3}{x}, \quad x \ne 0,\, x \ne 3
$$

4. Recognizing special patterns and structures

Many algebraic patterns (such as special products) are recognized and used through factoring.

Basic Types of Factors

When factoring polynomials in Algebra I, you will see several types of factors. It is helpful to recognize what kinds of factors are possible:

• Numeric factors
• Constants like $2$, $-5$, or $\frac{1}{3}$ that multiply the rest of the expression.
• Example: $12x^2 = 3 \cdot 4x^2$.
• Monomial factors
• Single-term expressions such as $x$, $3x^2$, or $-5xy$.
• Example: $8x^3y^2 = 4x^2y \cdot 2xy$.
• Binomial factors
• Two-term expressions such as $(x + 2)$ or $(3x - 5)$.
• Example: $x^2 - 4 = (x - 2)(x + 2)$.
• Trinomial factors
• Three-term expressions such as $(x^2 + x + 1)$.
• These often appear when factoring higher-degree polynomials:
  $$
  x^4 - 1 = (x^2 - 1)(x^2 + 1)
  $$

In typical Algebra I factoring problems, the final factors are usually numeric factors, monomials, or binomials.

General Strategy for Factoring

Factoring problems come in many forms, but the overall approach usually follows a pattern. A common strategy is:

1. Look for a greatest common factor (GCF)

Before doing anything else, check whether all terms share a common factor (numeric, variable, or both). Factor this out first.

Example:
$$
9x^3 - 6x^2 = 3x^2(3x - 2)
$$

2. Count the number of terms

After removing the GCF, look at how many terms remain:

• 2 terms ⇒ consider patterns like difference of squares or other special binomials.
• 3 terms ⇒ often a trinomial that may factor into two binomials.
• 4 or more terms ⇒ consider grouping or other methods.

3. Match to a known pattern or method
• For 2 terms: check for forms like $a^2 - b^2$.
• For 3 terms: check if it looks like a quadratic in standard form.
• For 4 terms: try grouping terms into pairs and factor each pair.
4. Check if factors can be factored further

After factoring once, look at each factor to see if it can be factored again.

• Example:
  $$
  x^4 - 16 = (x^2 - 4)(x^2 + 4) = (x - 2)(x + 2)(x^2 + 4)
  $$

5. Verify by multiplication (optional but useful)

Multiply your factors back out (mentally or on paper) to ensure you get the original expression.

Factoring by Grouping (Idea)

One general technique, especially for expressions with four terms, is factoring by grouping. The idea is to:

• Group terms into smaller parts,
• Factor each group,
• Then factor out what is common between the groups.

For instance, suppose we have:
$$
x^3 + 3x^2 + 2x + 6
$$

Group as:
$$
(x^3 + 3x^2) + (2x + 6)
$$

Factor each group:
$$
x^2(x + 3) + 2(x + 3)
$$

Now see that $(x + 3)$ is a common binomial factor:
$$
x^2(x + 3) + 2(x + 3) = (x^2 + 2)(x + 3)
$$

The key idea is to rearrange and group terms so that a common factor appears.

Even some trinomials can sometimes be handled by grouping, by rewriting the middle term as two terms.

Recognizing When a Polynomial Is Not Factorable (Over the Integers)

Not every polynomial can be factored nicely with integer coefficients. In Algebra I, we usually focus on factoring over the integers. A polynomial that cannot be factored into lower-degree polynomials with integer coefficients (other than taking out a common factor) is often called prime or irreducible over the integers.

Example:

• $x^2 + 5x + 6$ factors: $(x + 2)(x + 3)$.
• $x^2 + x + 1$ does not factor using integer coefficients, so it is prime in this context.

Signs that a quadratic $ax^2 + bx + c$ might be prime (over the integers):

• There are no two integers that multiply to $ac$ and add to $b$ (for the standard trinomial factoring methods).
• After trying reasonable factor combinations, no factoring works.

In later chapters, even “prime” quadratics can be factored over other number systems (for example, using complex numbers), but in Algebra I we typically stop once no integer-factorization is possible.

Common Pitfalls in Factoring

Several mistakes tend to appear when you first learn factoring:

1. Forgetting the greatest common factor

Example:

• Incorrect: $4x^2 + 8x = (x + 2)(4x)$
• Correct: $4x^2 + 8x = 4x(x + 2)$

Always check for a GCF first.

2. Dropping a factor

When factoring, every factor that appears must be kept. For example:

• Incorrect: $6x^2 - 12x = 6x - 12$
• Correct: $6x^2 - 12x = 6x(x - 2)$

3. Changing the value of the expression

Factoring should not change the value of the expression. For instance:

• Incorrect: $x^2 - 4x \to x(x - 4) - 4$
• The right side is not equal to the left side; a correct factorization is:
  $$
  x^2 - 4x = x(x - 4)
  $$

4. Leaving a common factor in all terms of a factor

If a factor still has a common factor within it, the expression is not fully factored.

Example:

• $6x^2 + 12x = 2x(3x + 6)$ is factored but not fully.
• Fully factored: $6x^2 + 12x = 6x(x + 2)$.

5. Sign errors

Sign mistakes are common, especially with negative terms or when factoring out a negative.

Example of factoring out a negative:
$$
-3x^2 + 6x = -3x(x - 2)
$$
Notice that $-3x(x - 2) = -3x^2 + 6x$.

Checking Your Factoring

A reliable way to confirm whether you factored correctly is to multiply the factors to see if you return to the original polynomial.

Example:

1. Suppose you claim that:
   $$
   x^2 - x - 6 = (x - 3)(x + 2)
   $$
2. Multiply:
   $$
   (x - 3)(x + 2) = x^2 + 2x - 3x - 6 = x^2 - x - 6
   $$
3. Since the expanded product matches the original polynomial exactly, the factorization is correct.

If the multiplication does not give back the original expression, then something in the factoring process needs to be corrected.

Practice Ideas

To become comfortable with factoring, practice problems with increasing variety and complexity, for example:

• Expressions with a simple common factor:
• $5x^2 + 10x$
• $12y^3 - 8y$
• Four-term polynomials suitable for grouping:
• $x^3 + x^2 + x + 1$
• $2x^3 + 4x^2 + 3x + 6$
• Expressions that might be prime over the integers so you learn to recognize when factoring stops.

In the upcoming subchapters, you will focus on specific types of factoring—such as factoring out common factors and factoring quadratics—and learn systematic procedures for them.