Factorization & Simplification
A complete, test-focused breakdown of every Factorization & Simplification concept on the ACT — from pulling out a GCF to factoring trinomials, recognizing special patterns, simplifying complex algebraic expressions, and applying these skills to solve equations.
In This Lesson
- 01 What Is Factoring & Why It Matters
- 02 Factoring Out the GCF
- 03 Difference of Squares
- 04 Perfect Square Trinomials
- 05 Factoring Trinomials (a = 1)
- 06 Factoring Trinomials (a ≠ 1)
- 07 Factoring by Grouping
- 08 Sum & Difference of Cubes
- 09 Simplifying Algebraic Expressions
- 10 Simplifying Rational Expressions
- 11 Solving Equations by Factoring
- 12 Test Day Strategy
- → Practice Quizzes
What Is Factoring & Why It Matters
Factoring is the process of rewriting an expression as a product of simpler expressions (its factors). It is the algebraic equivalent of reverse-multiplication — you are un-distributing or un-multiplying.
Multiplying factors → expanded form
Expanded form → product of factors
Factoring unlocks four major ACT skills:
1. Solving quadratic and polynomial equations (set each factor = 0).
2. Simplifying rational expressions (cancel common factors).
3. Evaluating expressions by recognizing a pattern and substituting.
4. Identifying zeros, roots, and x-intercepts of functions.
The Master Factoring Checklist
Every time you see a polynomial to factor on the ACT, run through these steps in order:
Always extract a GCF first — it simplifies every subsequent step. Then identify the structure of what remains.
Factoring Out the GCF
The first step in every factoring problem is to look for a Greatest Common Factor shared by all terms. Pull it out front using the distributive property in reverse.
// Think: what divides evenly into every term — coefficients AND variables
Finding the GCF of a Polynomial
- Numeric GCF: Largest integer dividing all coefficients.
- Variable GCF: Lowest power of each variable that appears in every term.
- Combined GCF: Multiply numeric GCF × variable GCF.
Don't skip the GCF step! Missing a GCF at the start leads to much harder factoring later. If the leading coefficient in a trinomial is large, always check for a numeric GCF first — it may reduce a complex problem to a simple one.
Difference of Squares
The difference of squares is the most frequently tested factoring pattern on the ACT. It applies whenever you have two perfect-square terms being subtracted.
// Note: a² + b² does NOT factor over the reals — sum of squares is PRIME
✓ Exactly two terms
✓ Minus sign between them
✓ Both terms are perfect squares (coefficients and variable powers are both perfect squares)
Always factor completely. After one round, check whether any resulting factor can be factored again. x⁴ − 81 requires two rounds. The ACT will include the intermediate (incomplete) factorization as a trap answer choice.
Perfect Square Trinomials
A perfect square trinomial is the expansion of a squared binomial. Recognizing it lets you factor in one step instead of working through the full trinomial method.
a² − 2ab + b² = (a − b)²
First term is a perfect square → a²
Last term is a perfect square → b²
Middle term = exactly ±2ab (twice the product of the square roots of the first and last terms)
Verify the middle term. Students often see two perfect square terms and assume it's a PST — but the middle term must be exactly 2ab. Always check all three conditions before concluding.
Factoring Trinomials When a = 1
The most common factoring task on the ACT: given x² + bx + c, find two binomials (x + p)(x + q) such that p + q = b and p × q = c.
where: p + q = b AND p × q = c
The "Product-Sum" Method
Find two numbers that multiply to c and add to b. List factor pairs of c and check which pair sums to b.
Sign Rules Summary
| c (last term) | b (middle term) | Both factors are… | Example |
|---|---|---|---|
| Positive | Positive | Both positive | x²+5x+6 → (+2)(+3) |
| Positive | Negative | Both negative | x²−5x+6 → (−2)(−3) |
| Negative | Positive | Larger factor positive | x²+x−6 → (+3)(−2) |
| Negative | Negative | Larger factor negative | x²−x−6 → (−3)(+2) |
Verify by expanding. After factoring, quickly FOIL your answer: (x + p)(x + q) = x² + (p+q)x + pq. If you get back the original trinomial, you're correct. This 5-second check saves you from trap answers on the ACT.
Factoring Trinomials When a ≠ 1
When the leading coefficient isn't 1, you have ax² + bx + c with a ≠ 1. Two reliable methods: the AC Method (factoring by splitting the middle term) and Trial and Error.
Method 1 — The AC Method
Method 2 — Trial and Error
Write (dx + e)(fx + g) where d×f = a and e×g = c. Test combinations until the outer+inner products sum to b.
Always pull GCF first. If 2x² + 7x + 6 had been 4x² + 14x + 12, pulling out the GCF of 2 first gives 2(2x² + 7x + 6) — then factor the simpler trinomial. Missing the GCF forces you to work with larger numbers unnecessarily.
Factoring by Grouping
When a polynomial has four terms, factoring by grouping is usually the method. Pair the first two and last two terms, factor each pair separately, then factor out the common binomial.
If grouping doesn't work: Try a different pairing (first + third, second + fourth) or rearrange the terms. Sometimes you need to factor out −1 from one group to make the binomials match: factor (a − b) from one group and −(b − a) = (a − b) from the other.
Sum & Difference of Cubes
These patterns apply to binomials where both terms are perfect cubes. They appear less frequently than difference of squares but do show up on harder ACT questions.
a³ − b³ = (a − b)(a² + ab + b²) ← Difference of cubes
The sign pattern for cube factoring: S O A P
Same sign as the original operation (+ or −)
Opposite sign in the trinomial's second term
Always Positive for the last term of the trinomial
Perfect Cubes to Memorize
6³=216 7³=343 8³=512 9³=729 10³=1000
Simplifying Algebraic Expressions
Simplification means rewriting an expression in its most compact equivalent form. On the ACT this includes combining like terms, applying exponent rules, and using the distributive property carefully.
Combining Like Terms
Terms are like terms if they have identical variable parts (same variable, same exponent). Add or subtract their coefficients only.
Distributive Property & Expanding
Distribute carefully — especially with negative signs, which must multiply every term inside the parentheses.
Exponent Rules for Simplification
| Rule | Form | Example |
|---|---|---|
| Product rule | xᵃ · xᵇ = xᵃ⁺ᵇ | x³ · x⁵ = x⁸ |
| Quotient rule | xᵃ / xᵇ = xᵃ⁻ᵇ | x⁷ / x² = x⁵ |
| Power rule | (xᵃ)ᵇ = xᵃᵇ | (x²)⁴ = x⁸ |
| Product to power | (xy)ᵃ = xᵃyᵃ | (2x)³ = 8x³ |
| Quotient to power | (x/y)ᵃ = xᵃ/yᵃ | (x/3)² = x²/9 |
| Negative exponent | x⁻ᵃ = 1/xᵃ | x⁻³ = 1/x³ |
| Zero exponent | x⁰ = 1 (x≠0) | (5x)⁰ = 1 |
| Fractional exponent | x^(m/n) = ⁿ√(xᵐ) | x^(3/2) = (√x)³ |
Simplifying Radical Expressions
√(a/b) = √a / √b ← Quotient under radical
√(a²) = |a| ← Perfect square out of radical
Simplifying Rational Expressions
A rational expression is a fraction with polynomials in the numerator and/or denominator. Simplifying one means factoring both and canceling common factors — never canceling terms.
Step 1: Factor the numerator completely.
Step 2: Factor the denominator completely.
Step 3: Cancel any factor that appears in both numerator and denominator. State restrictions (denominator ≠ 0).
Terms vs. Factors — the #1 ACT error:
ILLEGAL: (x + 5)/(x + 7) → 5/7 by "canceling x" ✗
LEGAL: x(x + 5) / [x(x + 7)] → (x + 5)/(x + 7) by canceling the factor x ✓
You can only cancel a factor that multiplies the entire numerator or the entire denominator.
Solving Equations by Factoring
Factoring is the primary tool for solving quadratic and polynomial equations on the ACT. The foundation is the Zero Product Property.
If A × B = 0, then A = 0 OR B = 0 (or both).
This is why factoring to solve equations works: once the polynomial equals zero, each factor can independently be set to zero to find solutions.
Standard 4-Step Process
Never divide both sides by a variable to "cancel" it. In 3x² = 12x, dividing both sides by x gives 3x = 12, so x = 4 only — you lose the solution x = 0! Always move everything to one side and factor instead.
Using Factoring to Find Zeros of a Function
The zeros (or roots) of f(x) are the x-values where f(x) = 0. They are the x-intercepts of the graph. Factor f(x) and set each factor to zero.
Test Day Strategy
Complete Factoring Pattern Reference
pq=c, p+q=b
AC method or trial & error
= a(x+y)+b(x+y)
= (a+b)(x+y)
Top Traps to Avoid
- Canceling terms, not factors: (x+3)/(x+5) ≠ 3/5.
- Forgetting x = 0 as a solution when a variable is a factor.
- Trying to factor a² + b² (sum of squares — it doesn't factor over reals).
- Not factoring completely — one more round may be possible.
- Skipping the GCF step and working with larger coefficients.
- Dividing both sides by a variable (loses a solution).
- Forgetting to state that the equation must equal zero before using Zero Product Property.
Speed Strategies
- Always factor GCF first — it simplifies every subsequent step.
- For x² + bx + c: use the sign rules table to know which sign combinations to test.
- Verify any factoring by quickly FOILing your answer.
- Plug in answer choices to check solutions — faster than re-solving on some problems.
- Spot difference of squares instantly: two terms, perfect squares, minus sign.
- For higher-degree polynomials, always look for GCF + pattern in what remains.
Put It to the Test
You've covered every factoring pattern and simplification technique. Now build speed and accuracy with ACT-style timed questions. Work through all five quizzes — each one targets different patterns and difficulty levels.