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Factoring Trinomials Calculator
Type a polynomial such as 6x² + 11x − 10 and get its complete factored form over the integers. The calculator pulls out the greatest common factor, recognises difference of squares, perfect squares and sums or differences of cubes, uses the ac method for trinomials and the rational root theorem for higher degrees, and explains every step.
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One variable, whole-number coefficients, powers with ^ (x^2). Up to degree 12.
Enter your values to see the result.
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How to use the Factoring Trinomials Calculator
- Type the polynomial using ^ for powers, e.g. 2x^3 - 8x (x² and x³ also work).
- Press Factor or just keep typing: the result updates live.
- Read the factored form and the method used.
- Follow the steps and check the roots table.
Factoring patterns
| Pattern | Formula | Example |
|---|---|---|
| Greatest common factor | ab + ac = a(b + c) | 6x² + 9x = 3x(2x + 3) |
| Difference of squares | A² − B² = (A − B)(A + B) | 9x² − 16 = (3x − 4)(3x + 4) |
| Perfect square trinomial | A² ± 2AB + B² = (A ± B)² | x² − 10x + 25 = (x − 5)² |
| Simple trinomial (a = 1) | x² + bx + c = (x + m)(x + n), mn = c, m + n = b | x² + 5x + 6 = (x + 2)(x + 3) |
| ac method (a ≠ 1) | split bx using mn = ac, m + n = b, then group | 2x² + 7x + 3 = (2x + 1)(x + 3) |
| 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) |
| Grouping (4 terms) | ax³ + bx² + cx + d, pair the terms | x³ + 3x² + 2x + 6 = (x + 3)(x² + 2) |
| Quadratic in form | substitute u = x² | x⁴ − 5x² + 4 = (x − 1)(x + 1)(x − 2)(x + 2) |
A factoring checklist
- Take out the greatest common factor (and a −1 if the leading coefficient is negative).
- Two terms: look for a difference of squares or a sum or difference of cubes.
- Three terms: check for a perfect square trinomial, then use the two-number or ac method.
- Four terms: try grouping.
- Degree 3 or more: test rational roots ±p/q and divide out each factor you find.
- Check every factor again until nothing more factors, then multiply back to verify.
Frequently asked questions
How do you factor a trinomial ax² + bx + c?
Multiply a × c, then find two numbers that multiply to ac and add to b. Split the middle term with those numbers and factor by grouping. For 6x² + 11x − 10: ac = −60, the numbers are 15 and −4, so 6x² + 15x − 4x − 10 = 3x(2x + 5) − 2(2x + 5) = (3x − 2)(2x + 5).
What should I always check first when factoring?
The greatest common factor. Take out any number and power of x that divides every term first: 2x³ − 8x = 2x(x² − 4) = 2x(x − 2)(x + 2).
How do I factor a difference of squares?
Use A² − B² = (A − B)(A + B). For 4x² − 25, A = 2x and B = 5, so it factors as (2x − 5)(2x + 5). A sum of squares like x² + 4 cannot be factored over the real numbers.
How do you factor a sum or difference of cubes?
A³ + B³ = (A + B)(A² − AB + B²) and A³ − B³ = (A − B)(A² + AB + B²). For example x³ + 27 = (x + 3)(x² − 3x + 9).
How do I know if a quadratic can be factored?
Compute the discriminant b² − 4ac. If it is a perfect square (0, 1, 4, 9, 16, …) the quadratic factors over the integers; otherwise it is prime over the integers, although it may still have irrational or complex roots.
How are cubic and higher polynomials factored?
The rational root theorem lists every possible rational root ±p/q, where p divides the constant term and q divides the leading coefficient. Each root r = p/q found gives a factor (qx − p); dividing it out leaves a smaller polynomial to keep factoring.
Can it factor expressions with two variables or decimals?
This tool works with one variable and whole-number coefficients. Multiply by a common denominator to clear fractions or decimals first, then factor.
Are all the tools on FreeToolCabin tested for quality and accuracy?
Yes! Every tool on our site undergoes automated Q/A testing in real browser environments to ensure high precision, correct output, and zero errors. If you happen to spot a bug or have a suggestion, please contact us at [email protected] and our team will review it right away.
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