Factor Polynomials & Trinomials
Enter any polynomial or trinomial to factor it step by step. This calculator handles GCF, difference of squares, trinomial factoring, AC method, and grouping.
Try These Examples
Click any example to load it into the calculator and see the step-by-step solution.
Factoring Methods Reference
Quick reference guide for the most common factoring patterns and when to use them.
| Method | Pattern | Factored Form | Example |
|---|---|---|---|
| GCF | ab + ac | a(b + c) | 6x + 9 = 3(2x + 3) |
| Difference of Squares | a² − b² | (a + b)(a − b) | x² − 9 = (x + 3)(x − 3) |
| Simple Trinomial | x² + bx + c | (x + m)(x + n) where mn = c, m+n = b | x² + 5x + 6 = (x + 2)(x + 3) |
| AC Method | ax² + bx + c | Split middle term, factor by grouping | 2x² + 7x + 3 = (2x + 1)(x + 3) |
| Factor by Grouping | ax + ay + bx + by | (a + b)(x + y) | x³ + 2x² + x + 2 = (x + 2)(x² + 1) |
| 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) |
How to Factor Polynomials
Factoring is the reverse of expanding. When you factor, you break a polynomial into smaller expressions that multiply to give the original. Here's the systematic approach:
Step 1: Always Factor Out the GCF First
The Greatest Common Factor (GCF) is the largest expression that divides all terms. Always check for it first.
Example: 6x² + 9x = 3x(2x + 3)
Both terms share 3x, so factor it out.
Step 2: Count the Terms
- Two terms: Look for difference of squares (a² − b²) or sum/difference of cubes
- Three terms (trinomials): Use trinomial factoring or AC method
- Four or more terms: Try factoring by grouping
Factoring Trinomials: x² + bx + c
When the leading coefficient is 1, find two numbers that:
- Multiply to give c (the constant term)
- Add to give b (the coefficient of x)
Example: x² + 7x + 12
- Need two numbers that multiply to 12 and add to 7
- 3 and 4 work: 3 × 4 = 12, 3 + 4 = 7
- Answer: (x + 3)(x + 4)
AC Method for ax² + bx + c (when a ≠ 1)
When the leading coefficient is not 1, use the AC method:
- Multiply a and c
- Find two numbers that multiply to ac and add to b
- Split the middle term using those two numbers
- Factor by grouping
Example: 2x² + 7x + 3
- a × c = 2 × 3 = 6
- Numbers that multiply to 6 and add to 7: 1 and 6
- Split middle term: 2x² + x + 6x + 3
- Group: x(2x + 1) + 3(2x + 1) = (x + 3)(2x + 1)
Difference of Squares
The pattern a² − b² always factors to (a + b)(a − b).
Examples:
- x² − 9 = (x + 3)(x − 3)
- 4x² − 25 = (2x + 5)(2x − 5)
- x² − 1 = (x + 1)(x − 1)
Note: The sum of squares (a² + b²) does NOT factor over real numbers.
Factor by Grouping
When you have four terms, group them in pairs and factor each pair:
Example: x³ + 2x² + 3x + 6
- Group: (x³ + 2x²) + (3x + 6)
- Factor each group: x²(x + 2) + 3(x + 2)
- Factor out common binomial: (x + 2)(x² + 3)
Common Factoring Mistakes
- Forgetting to check for GCF first. Always factor out the greatest common factor before using any other method.
- Sign errors. Pay close attention to positive and negative signs, especially when factoring trinomials with negative terms.
- Incomplete factoring. After factoring, check if any factors can be factored further. For example, x² − 4 = (x + 2)(x − 2), not just leaving it as x² − 4.
- Trying to factor sum of squares. Remember that a² + b² does not factor over real numbers (it's prime).
- Not checking your answer. Always expand your factored form to verify it matches the original expression.
Frequently Asked Questions
What is factoring in algebra?
Factoring is the process of breaking down a polynomial into simpler expressions (factors) that multiply together to give the original expression. For example, x² − 5x + 6 factors to (x − 2)(x − 3). It's the reverse of expanding or distributing.
What is the GCF method?
GCF (Greatest Common Factor) is the largest expression that divides all terms. Always factor out the GCF first before trying other methods. For example, 6x² + 9x = 3x(2x + 3) because 3x is the greatest common factor of both terms.
How do you factor a trinomial?
For x² + bx + c, find two numbers that multiply to c and add to b. For ax² + bx + c where a ≠ 1, use the AC method: multiply a and c, find factors of ac that add to b, split the middle term, then factor by grouping.
What is the difference of squares pattern?
The difference of squares pattern is a² − b² = (a + b)(a − b). For example, x² − 9 = (x + 3)(x − 3). This only works when you have subtraction between two perfect squares. The sum of squares (a² + b²) does NOT factor over real numbers.
What is the AC method for factoring?
The AC method factors ax² + bx + c by: (1) multiply a × c, (2) find two numbers that multiply to ac and add to b, (3) split the middle term using those numbers, (4) factor by grouping. This method works when the leading coefficient is not 1.
How do you factor by grouping?
Factor by grouping works on four-term polynomials: (1) group terms in pairs, (2) factor out the GCF from each pair, (3) factor out the common binomial. Example: ax + ay + bx + by = a(x + y) + b(x + y) = (a + b)(x + y).
Can all polynomials be factored?
Not all polynomials can be factored using integers or real numbers. When a polynomial cannot be factored, it is called prime or irreducible. For example, x² + x + 1 is prime over the integers (its discriminant is negative, so it has no real roots).
Does this calculator store my expressions?
No. All calculations happen entirely in your browser using JavaScript. Nothing is sent to a server or stored anywhere. Your data stays private on your device.
Privacy & Limitations
- All calculations run entirely in your browser -- nothing is sent to any server.
- Results are computed using standard formulas and should be verified for critical applications.
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Factoring Calculator FAQ
What is factoring in algebra?
Factoring is the process of breaking down a polynomial into simpler expressions (factors) that multiply together to give the original expression. For example, x² - 5x + 6 factors to (x - 2)(x - 3).
What is the GCF method?
GCF (Greatest Common Factor) is the largest expression that divides all terms. Always factor out the GCF first. For example, 6x² + 9x = 3x(2x + 3) because 3x is the greatest common factor.
How do you factor a trinomial?
For x² + bx + c, find two numbers that multiply to c and add to b. For ax² + bx + c where a ≠ 1, use the AC method: multiply a and c, find factors of ac that add to b, split the middle term, then factor by grouping.
What is the difference of squares pattern?
The difference of squares pattern is a² - b² = (a + b)(a - b). For example, x² - 9 = (x + 3)(x - 3). This only works when you have subtraction between two perfect squares.
What is the AC method for factoring?
The AC method factors ax² + bx + c by: (1) multiply a × c, (2) find two numbers that multiply to ac and add to b, (3) split the middle term using those numbers, (4) factor by grouping.
How do you factor by grouping?
Factor by grouping works on four-term polynomials: (1) group terms in pairs, (2) factor out the GCF from each pair, (3) factor out the common binomial. Example: ax + ay + bx + by = a(x + y) + b(x + y) = (a + b)(x + y).
Can all polynomials be factored?
Not all polynomials can be factored using integers. When a polynomial cannot be factored, it is called prime or irreducible. For example, x² + x + 1 is prime over the integers.
Does this calculator store my expressions?
No. All calculations happen in your browser. Nothing is sent to a server or stored anywhere.