The Timpson Lyceum · Intermediate Algebra (2023)

Lecture 12 (Intermediate Algebra) Factoring

Overview

Factoring is the reverse of multiplying out an expression: instead of distributing terms to expand a product, you start with the expanded expression and rewrite it as a product of factors. This lecture introduces the four core factoring techniques you will use throughout the chapter, in the order you should try them. It begins with the single most important first step — pulling out the greatest common factor (GCF) — then builds up through factoring by grouping, factoring trinomials with a leading coefficient of 1, and finally factoring general trinomials whose leading coefficient is not 1. Factoring is a tool: sometimes it simplifies an expression and sometimes it does not, but it is essential for solving equations later. The guiding rule across every problem is to look for a GCF first, then apply the appropriate method to what remains.

Key Concepts

Factoring as the reverse of distributing

• Multiplying takes a product like $4x^3(2y + 3x^2)$ and expands it to $8x^3y + 12x^5$.
• Factoring takes the expanded form $8x^3y + 12x^5$ and rewrites it back as the product $4x^3(2y + 3x^2)$.
• You can always check a factorization by multiplying it back out — you must get the original expression.

Greatest Common Factor (GCF)

• The GCF is the largest quantity that divides evenly into every term.
• Find it piece by piece: take the largest constant dividing all coefficients, then the smallest power of each variable that appears in every term.
• A variable that is missing from even one term contributes a power of zero — it cannot be part of the GCF. For example, the GCF of $24x^4y^2z,\ 18x^2y^4,\ 12x^3yz^5$ is $6x^2y$ (no $z$, since the middle term has none).
• Always pull out the GCF first, before trying any other method.

Factoring by grouping (four terms)

• Used when an expression has four terms. Group them in pairs and factor the GCF out of each pair.
• If the two pairs leave the same binomial inside parentheses, factor that common binomial out of the whole expression.
• Sometimes you must factor out a negative GCF from the second pair so the two binomials match.
• If the given order of terms does not produce a common binomial, rearrange the terms so that terms sharing factors are paired together.

Factoring trinomials with leading coefficient 1 ($x^2 + bx + c$)

• Look for two numbers whose product is $c$ (the constant) and whose sum is $b$ (the middle coefficient).
• The factorization is $(x + m)(x + n)$ where $m$ and $n$ are those two numbers.
• Sign rules: if $c>0$ the numbers share the sign of $b$; if $c<0$ one number is positive and one is negative.

Factoring general trinomials ($ax^2 + bx + c$, with $a \neq 1$)

• First always check for a GCF — pulling out a common factor may reduce $a$ to 1.
• If $a \neq 1$ remains, use the ac method: find two numbers whose product is $a\cdot c$ and whose sum is $b$.
• Split the middle term $bx$ into those two pieces, then factor by grouping.

Worked Example

Factor the general trinomial $3x^2 + 11x + 6$ using the ac method.

Check by expanding: $(3x+2)(x+3) = 3x^2 + 9x + 2x + 6 = 3x^2 + 11x + 6.$ ✓

Summary

• Factoring reverses distribution: rewrite an expanded expression as a product, and verify by multiplying back out.
• Always factor out the greatest common factor first — take the largest constant and the smallest power of each variable common to every term.
• For four-term expressions, group in pairs, factor each pair, then factor out the shared binomial; rearrange terms or factor out a negative when needed to make the binomials match.
• For $x^2 + bx + c$, find two numbers that multiply to $c$ and add to $b$, giving $(x+m)(x+n)$.
• For $ax^2 + bx + c$ with $a \neq 1$, use the ac method: find two numbers multiplying to $ac$ and adding to $b$, split the middle term, and factor by grouping.
• Use sign analysis (the signs of $b$ and $c$) to decide whether your two numbers are both positive, both negative, or one of each.