Factoring is one of the most useful skills in algebra. It lets you simplify expressions, solve equations and understand how a polynomial behaves. The idea is simple: rewrite a polynomial as a product of smaller pieces. This guide walks through the main methods in a sensible order, with worked examples and a quick checklist you can use on any problem.
What does it mean to factor?
To factor a polynomial means to write it as a product of other polynomials. For numbers, 12 = 3 × 4. For polynomials, x² + 5x + 6 = (x + 2)(x + 3). You can always check your work by multiplying the factors back together and confirming you get the original expression.
Step 1: Always look for a greatest common factor (GCF)
Before trying anything else, check whether every term shares a common factor. Take the largest number that divides all coefficients and the lowest power of each variable that appears in every term.
Example: Factor 6x³ + 9x².
The GCF of 6 and 9 is 3, and the lowest power of x is x². So the GCF is 3x².6x³ + 9x² = 3x²(2x + 3)
Step 2: Factoring by grouping
When a polynomial has four terms, try splitting it into two pairs, factoring a GCF out of each pair, and then pulling out the common binomial.
Example: Factor x³ + 2x² + 3x + 6.
Group: (x³ + 2x²) + (3x + 6)
Factor each group: x²(x + 2) + 3(x + 2)
Pull out the common binomial: (x + 2)(x² + 3)
Step 3: Factoring trinomials of the form x² + bx + c
Look for two numbers that multiply to c and add to b. Those numbers become the constants in the two binomials.
Example: Factor x² + 7x + 12.
We need two numbers with product 12 and sum 7: 3 and 4.x² + 7x + 12 = (x + 3)(x + 4)
Tip on signs: if c is positive, the two numbers have the same sign (both positive when b is positive, both negative when b is negative). If c is negative, the numbers have opposite signs. For instance, x² − 2x − 15 = (x − 5)(x + 3), because −5 × 3 = −15 and −5 + 3 = −2.
Step 4: Factoring trinomials of the form ax² + bx + c (a ≠ 1)
A reliable method is the “ac” method:
- Multiply a and c.
- Find two numbers that multiply to ac and add to b.
- Rewrite the middle term using those two numbers.
- Factor by grouping.
Example: Factor 2x² + 7x + 3.
ac = 6; the numbers that multiply to 6 and add to 7 are 1 and 6.
Rewrite: 2x² + x + 6x + 3
Group: x(2x + 1) + 3(2x + 1)
Result: (2x + 1)(x + 3)
Step 5: Special patterns worth memorizing
| Pattern | Formula | Example |
|---|---|---|
| Difference of squares | a² − b² = (a − b)(a + b) | x² − 25 = (x − 5)(x + 5) |
| Perfect square trinomial (plus) | a² + 2ab + b² = (a + b)² | x² + 6x + 9 = (x + 3)² |
| Perfect square trinomial (minus) | a² − 2ab + b² = (a − b)² | x² − 10x + 25 = (x − 5)² |
| Sum of cubes | a³ + b³ = (a + b)(a² − ab + b²) | x³ + 8 = (x + 2)(x² − 2x + 4) |
| Difference of cubes | a³ − b³ = (a − b)(a² + ab + b²) | x³ − 27 = (x − 3)(x² + 3x + 9) |
A common trap: the sum of squares, such as x² + 9, cannot be factored using real numbers. Do not confuse it with the difference of squares.
Step 6: Factor completely
After one round of factoring, check whether any of the new factors can be factored again. For example, 2x³ − 18x starts with a GCF of 2x: 2x(x² − 9). Then x² − 9 is a difference of squares, so the full answer is 2x(x − 3)(x + 3).
A quick checklist for any polynomial
- Factor out the GCF, including a negative sign if the leading term is negative and it helps.
- Count the terms: two terms suggest a special pattern; three terms suggest a trinomial method; four terms suggest grouping.
- Apply the matching method.
- Check whether any factor can be factored further.
- Multiply back to verify the result.
Why factoring matters: solving equations
Factoring is the key to the zero-product property: if a product equals zero, at least one factor must be zero. To solve x² − 5x + 6 = 0, factor to get (x − 2)(x − 3) = 0. Then set each factor equal to zero: x = 2 or x = 3. This approach works for many equations and also helps you simplify rational expressions by canceling common factors.
Common mistakes to avoid
- forgetting the GCF and ending up with a factorization that is not complete;
- mixing up signs when finding the two numbers for a trinomial;
- treating
a² + b²as if it factored like a difference of squares; - losing a factor of 1 when the whole term is pulled out, for example
4x + 4 = 4(x + 1), not4(x); - skipping the final multiplication check.
Practice problems to try
The best way to lock in these methods is to practice a little every day. Try factoring the following expressions on paper, then check each one by multiplying the factors back together:
10x² + 15x(hint: start with the GCF)x² + 9x + 20(hint: two numbers that multiply to 20 and add to 9)x² − 49(hint: difference of squares)3x² + 10x + 8(hint: use the “ac” method)x³ + 3x² + 4x + 12(hint: grouping)
Answers: (1) 5x(2x + 3); (2) (x + 4)(x + 5); (3) (x − 7)(x + 7); (4) (3x + 4)(x + 2); (5) (x + 3)(x² + 4). If your result does not match, look for the step where a sign or a coefficient slipped, which is the most common source of errors.
When you feel comfortable, mix the problem types together without hints. In an exam, the hardest part is often deciding which method to use, so practicing that decision is just as valuable as practicing the algebra itself. Keep a short notebook of the patterns you tend to forget, and review it before every quiz.
Conclusion
Factoring becomes easier once you follow a fixed order: GCF first, then choose the method based on the number of terms, and always factor completely. With regular practice, the patterns start to feel automatic.
If you want structured practice, take a look at the algebra and math courses on Cursa. They offer guided lessons and exercises to help you build confidence step by step.
