Algebra Made Clear
Factoring Polynomials: Techniques for Solving Equations
Turn a complicated-looking polynomial into simple factors—and find the solutions with confidence.
What Does Factoring Mean?
To factor a polynomial means to rewrite it as a product of simpler expressions. For example, the expression x2 + 5x + 6 factors to (x + 2)(x + 3) because multiplying those two binomials gives the original polynomial.
When solving an equation, the most useful target is a product equal to zero. The key rule is:
On a small phone screen, swipe horizontally inside this equation box if needed.
The Factoring Toolkit
Use this quick decision process before you start expanding or guessing.
| Look for | Technique | Example pattern |
|---|---|---|
| A common factor in every term | Greatest common factor (GCF) | 6x2 + 9x = 3x(2x + 3) |
| Three terms, leading coefficient 1 | Simple trinomial | x2 + 7x + 12 = (x + 3)(x + 4) |
| Two perfect squares separated by − | Difference of squares | a2 − b2 = (a − b)(a + b) |
| Four terms in pairs | Grouping | ax + ay + bx + by = (a + b)(x + y) |
Mobile note: Swipe left or right inside the table on a small phone screen to read every column.
Step-by-Step Worked Examples
Example 1: Factor out the GCF
Solve: 6x2 + 9x = 0
Step 1: The GCF of 6x2 and 9x is 3x.
Step 2: Factor it out: 3x(2x + 3) = 0.
Step 3: Set each factor equal to zero: 3x = 0 or 2x + 3 = 0.
Solutions: x = 0 or x = −3 ÷ 2.
Example 2: Factor a simple trinomial
Solve: x2 + 7x + 12 = 0
Find two numbers whose product is 12 and whose sum is 7. They are 3 and 4.
Therefore, x2 + 7x + 12 = (x + 3)(x + 4).
Now use the zero-product property: x + 3 = 0 or x + 4 = 0.
Solutions: x = −3 or x = −4.
Example 3: Difference of squares
Solve: 9x2 − 25 = 0
Recognize two squares: 9x2 = (3x)2 and 25 = 52.
Use the formula a2 − b2 = (a − b)(a + b):
So 3x − 5 = 0 or 3x + 5 = 0.
Solutions: x = 5 ÷ 3 or x = −5 ÷ 3.
Example 4: Factoring by grouping
Solve: x3 + 2x2 + 3x + 6 = 0
Group the first two and last two terms:
x2(x + 2) + 3(x + 2) = 0
Factor out the common binomial: (x + 2)(x2 + 3) = 0.
For real-number solutions, x + 2 = 0 gives x = −2. The equation x2 + 3 = 0 has no real solution because x2 cannot equal −3.
A Reliable Solving Routine
- Move every term to one side so the equation equals zero.
- Combine like terms and look for a greatest common factor.
- Choose the pattern: trinomial, difference of squares, grouping, or another method.
- Set each factor equal to zero.
- Check each answer in the original equation, not only in the factored equation.
Memory Tip
G-Z-F-S-C: Get to zero, find a common factor, spot the pattern, split the factors, and check. Say those five words before you begin and you will avoid many skipped steps.
Common Mistakes to Avoid
- Forgetting the equation must equal zero: The zero-product property does not apply directly to a product equal to 5 or 12.
- Missing the GCF: Always check every term first. A hidden common factor can make the rest of the problem much easier.
- Using the wrong signs: For x2 + bx + c, the two numbers must multiply to c and add to b.
- Stopping too soon: After factoring out a GCF, inspect the remaining factor for another pattern.
- Not checking: Substitute solutions into the original equation to catch sign and arithmetic errors.
Practice: Try These First
- Factor and solve: x2 + 9x + 20 = 0
- Factor and solve: 4x2 − 16 = 0
- Factor and solve: 5x2 − 15x = 0
- Factor and solve: x3 − 4x2 + 2x − 8 = 0
- Factor and solve: x2 − x − 12 = 0
Answer Section
1. (x + 4)(x + 5) = 0; x = −4 or x = −5.
2. 4(x − 2)(x + 2) = 0; x = 2 or x = −2.
3. 5x(x − 3) = 0; x = 0 or x = 3.
4. Grouping gives (x2 + 2)(x − 4) = 0; the real solution is x = 4.
5. (x − 4)(x + 3) = 0; x = 4 or x = −3.
Final Takeaway
Factoring is a pattern-recognition skill. Put the equation equal to zero, take out the GCF, choose the right pattern, apply the zero-product property, and check your solutions. With practice, many intimidating polynomial equations become a short sequence of manageable steps.
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