ACT Math: Factorization & Simplification
Factorization & Simplification is a core Algebra topic on the ACT, and it's the toolkit you'll reach for constantly, inside equation-solving, quadratics, and rational expression problems alike. This free, complete lesson covers combining like terms and the distributive property, adding, subtracting, and multiplying polynomials, factoring by GCF and by grouping, and factoring patterns like the difference of squares and perfect square trinomials, along with evaluating polynomials by substitution. Each idea is followed by a fully worked example, and then a set of original practice problems with instant feedback, in the same four-answer-choice, calculator-allowed format as the current ACT. Everything here is free, and you can keep practicing afterward with the full ACT Math Question Bank linked below.
1. Combining Like Terms and the Distributive Property
Like terms have the same variable(s) raised to the same power(s), and only like terms can be combined by adding or subtracting their coefficients. The distributive property, a(b + c) = ab + ac, lets you expand an expression before combining terms.
Worked Example: Simplify 3x + 2y − 5x + 7y − 4.
Group by type: x terms: 3x − 5x = −2x. y terms: 2y + 7y = 9y. Constant: −4.
= −2x + 9y − 4
1. Simplify: 5a − 3b + 2a + 8b
Explanation: a terms: 5a + 2a = 7a. b terms: −3b + 8b = 5b. Result: 7a + 5b.
2. Simplify: 4(2x − 3) + 5x
Explanation: Distribute: 8x − 12 + 5x. Combine like terms: 13x − 12.
3. Simplify: −2(3x + 4) − 5x
Explanation: Distribute: −6x − 8 − 5x. Combine like terms: −11x − 8.
4. Simplify: 6x² − 4x + 2x² + 7x
Explanation: x² terms: 6x² + 2x² = 8x². x terms: −4x + 7x = 3x. Result: 8x² + 3x.
5. Simplify: 3(x + 2y) − 2(x − y)
Explanation: Distribute: 3x + 6y − 2x + 2y. Combine like terms: (3x − 2x) + (6y + 2y) = x + 8y.
2. Adding, Subtracting, and Multiplying Polynomials
Add or subtract polynomials by combining like terms (remember to distribute a negative sign across every term when subtracting). Multiply two binomials using FOIL: First, Outer, Inner, Last.
Worked Example: Multiply (2x + 3)(x − 5).
2x(x) + 2x(−5) + 3(x) + 3(−5)
2x² − 10x + 3x − 15 = 2x² − 7x − 15
1. Add: (4x² + 6x − 7) + (2x² − 5x − 3)
Explanation: x²: 4x² + 2x² = 6x². x: 6x − 5x = x. Constants: −7 − 3 = −10. Result: 6x² + x − 10.
2. Subtract: (9x² − 2x + 4) − (10x² + 8x − 2)
Explanation: Distribute the minus sign: 9x² − 2x + 4 − 10x² − 8x + 2. Combine: (9x² − 10x²) + (−2x − 8x) + (4 + 2) = −x² − 10x + 6.
3. Multiply: (x + 4)(x − 6)
Explanation: x² − 6x + 4x − 24 = x² − 2x − 24.
4. Multiply: (3x − 2)(2x + 5)
Explanation: 6x² + 15x − 4x − 10 = 6x² + 11x − 10.
5. Multiply: 3x²(2x − 5)
Explanation: Distribute 3x² to each term: 3x²(2x) − 3x²(5) = 6x³ − 15x².
3. Factoring: GCF and Grouping
To factor out a greatest common factor (GCF), find the largest expression that divides every term, then pull it outside parentheses. For a four-term polynomial that doesn't share a single GCF across all terms, try factoring by grouping: split the polynomial into two pairs, factor each pair separately, then factor out the common binomial.
Worked Example: Factor 6x³ − 15x² completely.
The GCF of 6x³ and 15x² is 3x².
6x³ − 15x² = 3x²(2x − 5)
1. Factor completely: 8x² + 20x
Explanation: The GCF of 8x² and 20x is 4x: 8x² + 20x = 4x(2x + 5).
2. Factor completely: 12x³ − 18x²
Explanation: The GCF of 12x³ and 18x² is 6x²: 12x³ − 18x² = 6x²(2x − 3).
3. Factor by grouping: x³ + 3x² + 2x + 6
Explanation: Group: (x³ + 3x²) + (2x + 6) = x²(x + 3) + 2(x + 3). Factor out the common binomial: (x + 3)(x² + 2).
4. Factor by grouping: 2x³ − 6x² + 5x − 15
Explanation: Group: (2x³ − 6x²) + (5x − 15) = 2x²(x − 3) + 5(x − 3). Factor out the common binomial: (x − 3)(2x² + 5).
5. What is the greatest common factor of 6x²y and 9xy²?
Explanation: The numeric GCF of 6 and 9 is 3. Take the lowest power of each shared variable: x¹ (from x and x²) and y¹ (from y and y²). GCF = 3xy.
4. Factoring Patterns and Evaluating Polynomials
Two patterns are worth recognizing on sight: the difference of squares, a² − b² = (a − b)(a + b), and the perfect square trinomial, a² ± 2ab + b² = (a ± b)². To evaluate a polynomial at a given value, substitute that value for the variable and simplify using order of operations.
Worked Example: Factor x² − 49.
This matches a² − b² with a = x and b = 7.
x² − 49 = (x − 7)(x + 7)
1. Factor: x² − 81
Explanation: x² − 81 = x² − 9² = (x − 9)(x + 9).
2. Factor: 4x² − 25
Explanation: 4x² − 25 = (2x)² − 5² = (2x − 5)(2x + 5).
3. Factor: x² + 10x + 25
Explanation: This matches a² + 2ab + b² with a = x and b = 5, since 2(x)(5) = 10x, so it factors as (x + 5)².
4. Evaluate the polynomial p(x) = 2x² − 3x + 5 at x = −2.
Explanation: p(−2) = 2(−2)² − 3(−2) + 5 = 2(4) + 6 + 5 = 8 + 6 + 5 = 19.
5. Evaluate the polynomial p(x) = x³ − 4x + 1 at x = 3.
Explanation: p(3) = 3³ − 4(3) + 1 = 27 − 12 + 1 = 16.
Common Mistakes to Avoid
- Forgetting to distribute a negative sign across every term when subtracting polynomials. (9x² − 2x + 4) − (10x² + 8x − 2) requires flipping the sign of all three terms in the second polynomial, not just the first one.
- Combining terms that aren't actually like terms. x² and x are different powers of the same variable and cannot be combined, even though they share the variable x.
- Stopping after pulling out a partial GCF. If every remaining term inside the parentheses still shares a common factor, the expression isn't fully factored yet.
- Misapplying the perfect square trinomial pattern. x² + 25 is not a perfect square trinomial and does not factor as (x + 5)², that pattern only applies when the middle term matches 2ab exactly.
- Making an arithmetic slip when evaluating a polynomial. Substituting a negative value is especially error-prone, always work through order of operations carefully, particularly when squaring or cubing a negative number.
Frequently Asked Questions
How do I know whether to factor by GCF, grouping, or a special pattern?
Always check for a GCF first, even if you'll use another method afterward, since factoring out a GCF simplifies the remaining work. For a four-term polynomial with no overall GCF, try grouping. For a two-term or three-term expression, check whether it matches a difference of squares or perfect square trinomial pattern before falling back to general methods.
Is factoring the same as solving an equation?
No. Factoring rewrites an expression as a product, it doesn't require the expression to equal anything. Solving a quadratic equation is a related but separate skill that uses factoring as one possible tool, once you set the expression equal to zero.
Why does evaluating polynomials matter on the ACT?
Substitution questions test whether you can apply order of operations accurately under time pressure, and they often appear as a quick way to check whether a proposed factoring or simplification is correct, by plugging in a convenient number and comparing results.
Where can I practice more problems like these?
The Factorization & Simplification quizzes in the ACT Math Question Bank include additional original problems on this topic, along with quizzes covering every other topic tested on the ACT Math section.