Problems to Factor- Practice with Solutions
What Is Factoring and Why It Matters
Factoring is breaking down complex expressions into simpler parts that multiply together to give you the original expression. It's the reverse of distribution. If you've ever distributed 4(x + 2) to get 4x + 8, factoring is doing the opposite—turning 4x + 8 back into 4(x + 2).
You need this skill for solving equations, simplifying fractions, and eventually for algebra, calculus, and beyond. If you can't factor reliably, you'll hit a wall in math class fast.
Types of Factoring Problems You'll Encounter
Not all factoring is the same. The method you use depends on the structure of the expression.
Greatest Common Factor (GCF)
This is the simplest type. Find the largest number or term that divides every part of the expression.
Example: 12x³ + 18x²
The GCF of 12 and 18 is 6. The GCF of x³ and x² is x². So the GCF is 6x².
Answer: 6x²(2x + 3)
Factoring Trinomials
These expressions have three terms and look like ax² + bx + c. You need to find two binomials that multiply to give you the original.
Example: x² + 5x + 6
You need two numbers that multiply to 6 and add to 5. Those numbers are 2 and 3.
Answer: (x + 2)(x + 3)
Difference of Squares
When you have something minus something squared, it factors into two binomials with opposite signs.
Pattern: a² - b² = (a + b)(a - b)
Example: x² - 16
This is x² - 4², so it factors to (x + 4)(x - 4).
Perfect Square Trinomials
These are trinomials that come from squaring a binomial.
Pattern: a² + 2ab + b² = (a + b)²
Example: x² + 6x + 9
This is (x + 3)² because x² + 2(3)x + 9.
Factoring by Grouping
Used when you have four or more terms. Group terms strategically, factor out GCF from each group, then look for a common binomial factor.
Practice Problems with Solutions
Work through these. Try each one before checking the answer.
GCF Problems
1. 8y + 12
GCF of 8 and 12 is 4.
Answer: 4(2y + 3)
2. 15a²b - 25ab²
GCF is 5ab.
Answer: 5ab(3a - 5b)
3. 6m³ + 9m² - 3m
GCF is 3m.
Answer: 3m(2m² + 3m - 1)
Trinomial Problems
4. x² + 7x + 12
Two numbers that multiply to 12 and add to 7: 3 and 4.
Answer: (x + 3)(x + 4)
5. x² - 4x - 12
Two numbers that multiply to -12 and add to -4: -6 and 2.
Answer: (x - 6)(x + 2)
6. 2x² + 7x + 3
This one needs trial and error or the AC method. Numbers that multiply to 6 (2×3) and add to 7: 6 and 1.
Rewrite: 2x² + 6x + x + 3
Group: (2x² + 6x) + (x + 3)
Factor: 2x(x + 3) + 1(x + 3)
Answer: (2x + 1)(x + 3)
Difference of Squares
7. 4x² - 25
This is (2x)² - 5².
Answer: (2x + 5)(2x - 5)
8. 9y² - 64
This is (3y)² - 8².
Answer: (3y + 8)(3y - 8)
9. x⁴ - 16
This is (x²)² - 4². First: (x² + 4)(x² - 4). Then factor x² - 4 as (x + 2)(x - 2).
Answer: (x² + 4)(x + 2)(x - 2)
Perfect Square Trinomials
10. x² + 10x + 25
Check: is 2ab equal to 10x? x² + 2(x)(5) + 25. Yes.
Answer: (x + 5)²
11. 4x² - 12x + 9
This is (2x)² - 2(2x)(3) + 3².
Answer: (2x - 3)²
Factoring by Grouping
12. x³ + 2x² + 3x + 6
Group: (x³ + 2x²) + (3x + 6)
Factor each: x²(x + 2) + 3(x + 2)
Answer: (x² + 3)(x + 2)
13. 2ax + 2ay + bx + by
Group: (2ax + 2ay) + (bx + by)
Factor: 2a(x + y) + b(x + y)
Answer: (2a + b)(x + y)
Factoring Methods Comparison
| Method | Best Used When | Pattern to Recognize |
|---|---|---|
| GCF | All terms share a common factor | Numbers or variables appear in every term |
| Trinomial Factoring | Expression has three terms, starts with x² | ax² + bx + c form |
| Difference of Squares | Two perfect squares subtracted | a² - b² |
| Perfect Square Trinomial | First and last terms are perfect squares | a² ± 2ab + b² |
| Factoring by Grouping | Four or more terms | Can split into groups with common factors |
How to Factor Any Expression: Step-by-Step
Follow this order. Don't skip steps or you'll miss easier factorizations.
Step 1: Factor out the GCF first
Always check for a common factor before anything else. This simplifies everything.
Step 2: Count the terms
- Two terms → check for difference of squares (a² - b²)
- Three terms → check if it's a perfect square trinomial, otherwise use trinomial factoring
- Four terms → try factoring by grouping
Step 3: Check your work
Multiply the factors back out. You should get the original expression. This takes two seconds and catches most mistakes.
Common Mistakes to Avoid
Forgetting the GCF is the biggest error students make. They try fancy methods on expressions that simplify first. Always check for a common factor.
Getting the signs wrong in trinomials. If the constant is negative, one binomial gets a negative sign, not both. If you see x² - 5x + 6, the factors are (x - 2)(x - 3) because -2 × -3 = +6.
Not checking your answer. Factoring is not complete until you verify by multiplying. Do this every time until it becomes habit.
Overcomplicating things. Some students see a four-term expression and jump to grouping without noticing a simpler GCF first.