AI Factoring Calculator
Factoring turns a sum into a product, for example x² + 7x + 12 into (x + 3)(x + 4). This AI factoring calculator finds the factors and explains the pattern behind them.
The factoring checklist
- Take out the greatest common factor first. 6x² + 9x becomes 3x(2x + 3).
- Look for a special pattern: difference of squares, perfect-square trinomials, sum or difference of cubes.
- For x² + bx + c, find two numbers that multiply to c and add to b.
- For ax² + bx + c, multiply a × c, find the pair that adds to b, and split the middle term.
Special patterns worth remembering
| Pattern | Factored form |
|---|---|
| a² − b² | (a − b)(a + b) |
| a² + 2ab + b² | (a + b)² |
| a² − 2ab + b² | (a − b)² |
| a³ − b³ | (a − b)(a² + ab + b²) |
| a³ + b³ | (a + b)(a² − ab + b²) |
Factoring is the fastest way to solve many quadratics. Set each bracket to zero, or compare with the formula in the quadratic equation solver.
Check by expanding
You can always verify a factorisation by multiplying the brackets back out. If you recover the original expression, it is right. The algebra solver can expand brackets for you.
Worked examples
Example 1: Factor x² + 7x + 12
- Find two numbers with product 12 and sum 7: 3 and 4.
- Write the brackets.
Final answer: (x + 3)(x + 4)
Example 2: Factor x² − 9
- This is a difference of squares: x² − 3².
- Apply a² − b² = (a − b)(a + b).
Final answer: (x − 3)(x + 3)
Example 3: Factor 2x² + 7x + 3
- Multiply a × c: 2 × 3 = 6. Find a pair adding to 7: 6 and 1.
- Split the middle: 2x² + 6x + x + 3.
- Group: 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3).
Final answer: (2x + 1)(x + 3)
Example 4: Factor x³ − 8
- Write as x³ − 2³, a difference of cubes.
- Apply (a − b)(a² + ab + b²) with a = x and b = 2.
Final answer: (x − 2)(x² + 2x + 4)
Example 5: Factor 6x² + 9x
- The greatest common factor of 6x² and 9x is 3x.
- Divide each term by 3x.
Final answer: 3x(2x + 3)
Common mistakes to avoid
- Missing a common factor that makes the rest easier.
- Choosing a pair that multiplies correctly but adds to the wrong sum.
- Mixing up signs when the constant is negative.
- Trying to factor a sum of squares such as x² + 9 over the real numbers.
- Not checking by expansion.
Frequently asked questions
What if the expression cannot be factored?
Some quadratics do not factor with whole numbers. The discriminant tells you; use the quadratic formula instead.
Can the calculator factor cubic polynomials?
Yes, especially those that fit a pattern or have a rational root. AI mode explains the approach.
What is the greatest common factor?
The largest term that divides every term in the expression.
Is factoring the same as solving?
No. Factoring rewrites an expression; solving finds the values that make an equation true. Factoring is often step one of solving.
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