Factoring Polynomials- Techniques and Examples
What Factoring Polynomials Actually Is
Factoring is breaking down a polynomial into simpler pieces that multiply together to give you the original expression. That's it. No magic, no philosophy.
You need this skill because it simplifies equations, solves problems faster, and shows up constantly in algebra, calculus, and beyond. If you're still solving quadratic equations by brute force, you're wasting time.
The Core Techniques You Must Know
1. Factoring Out the Greatest Common Factor (GCF)
This is always your first move. Look at every term and find what they share.
Example:
12x³ + 18x²
GCF is 6x². Pull it out:
6x²(2x + 3)
Done. That's the entire process. Find the biggest number that divides all terms, find the biggest variable power that appears in every term, factor it out front.
2. Factoring by Grouping
Use this when you have four terms with no obvious GCF across all of them.
Example:
3x³ + 2x² - 6x - 4
Group the first two and last two:
(3x³ + 2x²) + (-6x - 4)
Factor each group:
x²(3x + 2) - 2(3x + 2)
Now you see the common binomial (3x + 2):
(3x + 2)(x² - 2)
Doesn't always work. If it doesn't, try a different grouping or move to another technique.
3. Factoring Trinomials
The standard form is ax² + bx + c. You're looking for two binomials that multiply to give you this.
When a = 1:
x² + 5x + 6
Find two numbers that multiply to 6 (the constant) and add to 5. That's 2 and 3.
(x + 2)(x + 3)
When a ≠ 1:
2x² + 7x + 3
Multiply a and c: 2 × 3 = 6. Find two numbers that multiply to 6 and add to 7. That's 6 and 1.
Rewrite: 2x² + 6x + x + 3
Group: (2x² + 6x) + (x + 3)
Factor: 2x(x + 3) + 1(x + 3)
Final: (2x + 1)(x + 3)
The AC method (multiply a and c, find factors that sum to b) works every time. Some people prefer the "guess and check" approach with FOIL, but that's slower and error-prone.
4. Difference of Squares
This pattern only applies to expressions that subtract one perfect square from another.
The formula:
a² - b² = (a + b)(a - b)
Example:
x² - 16
This is x² - 4². Apply the formula:
(x + 4)(x - 4)
Works with variables too: 49y² - 25 = (7y + 5)(7y - 5)
You cannot factor a sum of squares using real numbers. x² + 16 doesn't factor over the reals. Don't try.
5. Perfect Square Trinomials
Recognize these two patterns:
a² + 2ab + b² = (a + b)²
a² - 2ab + b² = (a - b)²
Example:
x² + 10x + 25
Check: x² is a², 25 is b² (5²), and 10x is 2ab where 2(5) = 10. It fits.
(x + 5)²
6. Sum and Difference of Cubes
Less common but you need them for harder problems.
Formulas:
a³ + b³ = (a + b)(a² - ab + b²)
a³ - b³ = (a - b)(a² + ab + b²)
Example:
x³ - 27
This is x³ - 3³.
(x - 3)(x² + 3x + 9)
The trinomial part never factors further over the reals. Remember that.
Factoring Techniques Comparison
| Technique | Best Used When | Pattern/Formula |
|---|---|---|
| GCF | Always check first | Factor out common term |
| Grouping | 4 terms, no common GCF | Group in pairs |
| Trinomials (a=1) | x² + bx + c | Find factors of c that sum to b |
| Trinomials (a≠1) | ax² + bx + c | AC method |
| Difference of Squares | a² - b² | (a+b)(a-b) |
| Perfect Square Trinomial | a² ± 2ab + b² | (a±b)² |
| Cubes | a³ ± b³ | Cube formulas above |
How to Factor Any Polynomial
Follow this decision tree every time. No exceptions.
- Step 1: Check for a GCF across all terms. Factor it out first.
- Step 2: Count your terms.
- 2 terms? Check for difference of squares or cubes.
- 3 terms? Check if it's a perfect square, otherwise use trinomial factoring.
- 4 terms? Try grouping.
- Step 3: Check if any binomial factors can be factored again. Keep pulling until you can't anymore.
- Step 4: Verify by multiplying back with FOIL.
Example walkthrough:
Factor: 2x³ - 8x
Step 1: GCF is 2x. 2x(x² - 4)
Step 2: x² - 4 is a difference of squares. Factor: 2x(x + 2)(x - 2)
Step 3: None of these factor further. Done.
Common Mistakes That Cost You Points
- Forgetting to check for GCF before trying other methods. This is the biggest time-waster.
- Misidentifying a + b² as a difference of squares. It isn't. Only a² - b² fits.
- Dropping signs when grouping. Watch those negative signs like a hawk.
- Not checking your work. Multiply the factors back. If you don't get the original, you failed.
- Trying to factor a prime polynomial. Some things don't factor. Move on.
Quick Reference for Standard Forms
Memorize these patterns. They come up constantly:
- x² - y² = (x + y)(x - y)
- x² + 2xy + y² = (x + y)²
- x² - 2xy + y² = (x - y)²
- x³ + y³ = (x + y)(x² - xy + y²)
- x³ - y³ = (x - y)(x² + xy + y²)
That's everything you need. Practice the examples, memorize the formulas, and check your work every single time. Factoring is a mechanical skill—repetition is the only way to get fast at it.