Factoring Calculator (Polynomials & Trinomials, Free)
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Factoring rewrites a polynomial as a product of simpler polynomials. Take out the greatest common factor first, then look for a pattern. The factoring calculator names each technique it uses. For example, 2x³ + 3x² − 8x − 12 groups as x²(2x + 3) − 4(2x + 3) = (x² − 4)(2x + 3), and the difference of squares finishes it: (x − 2)(x + 2)(2x + 3).
Show steps (4)
- Greatest common factor Always check for a GCF first. Here the terms have no common factor other than 1, so move on to the patterns.
- Grouping Group the four terms in pairs: (2x3 + 3x2) + (−8x − 12). Take the GCF out of each pair: x2(2x + 3) − 4(2x + 3). Both parts share (2x + 3), so factor it out: 2x3 + 3x2 − 8x − 12 = (x2 − 4)(2x + 3).
- Difference of squares Both terms are perfect squares with a minus sign between them: x2 − 4 = x2 − 22. Use a2 − b2 = (a − b)(a + b) with a = x and b = 2: x2 − 4 = (x − 2)(x + 2).
- Check Multiply the factors back together: (x − 2)(x + 2)(2x + 3) = 2x3 + 3x2 − 8x − 12 ✓ This matches 2x3 + 3x2 − 8x − 12, so the factored form is correct.
How to use the factoring calculator
- Type a polynomial in the Polynomial tab, for example 6x^2 + 11x – 10. Use ^ for powers.
- To factor a quadratic trinomial from its coefficients, open the Trinomial tab and enter a, b and c.
- To factor a number such as 360, use the Whole number tab.
- Read the factored form, then open Show steps. Each step names the technique it uses, and the last step checks the answer by multiplying the factors back together.
Which factoring technique to use
Work down this list, then repeat it on every factor that is left.
| Technique | How to spot it | Example |
|---|---|---|
| Greatest common factor (GCF) | Every term shares a number or a power of x | 6x³ − 24x = 6x(x² − 4) |
| Difference of squares | Two squares with a minus between them | x² − 25 = (x − 5)(x + 5) |
| Sum or difference of cubes | Two cubes such as x³ and 27 | x³ + 27 = (x + 3)(x² − 3x + 9) |
| Perfect square trinomial | First and last terms are squares, middle term is 2ab | 4x² − 12x + 9 = (2x − 3)² |
| Trinomial with a = 1 | x² + bx + c | x² + 7x + 12 = (x + 3)(x + 4) |
| AC method | ax² + bx + c with a ≠ 1 | 2x² + 7x + 3 = (2x + 1)(x + 3) |
| Grouping | Four terms | x³ + 2x² + 3x + 6 = (x² + 3)(x + 2) |
| Rational Root Theorem | Degree 3 or higher with no pattern | x³ − 6x² + 11x − 6 = (x − 1)(x − 2)(x − 3) |
Factoring trinomials with the ac-method
For x² + bx + c, find two numbers that multiply to c and add to b. In x² + 7x + 12, the numbers 3 and 4 multiply to 12 and add to 7, so x² + 7x + 12 = (x + 3)(x + 4).
When the leading coefficient a is not 1, use the ac-method:
- Multiply a by c.
- Find two numbers that multiply to ac and add to b.
- Use those two numbers to split the middle term.
- Group the four terms in pairs and factor out the common binomial.
Worked example: 2x² + 7x + 3
Here ac = 2 · 3 = 6. The pair 1 and 6 multiplies to 6 and adds to 7. Split the middle term: 2x² + x + 6x + 3. Group: x(2x + 1) + 3(2x + 1) = (2x + 1)(x + 3).
Worked example: 6x² + 11x − 10
Here ac = 6 · (−10) = −60. List factor pairs of −60 until the sum is 11: −4 and 15 work. Split: 6x² + 15x − 4x − 10. Group: 3x(2x + 5) − 2(2x + 5) = (3x − 2)(2x + 5).
Factoring completely: take out the GCF first
A polynomial is factored completely when no factor can be broken down further. Start with the GCF, which makes the numbers smaller. In 3x² − 12x − 15 every coefficient is a multiple of 3: 3(x² − 4x − 5). Then −5 and 1 multiply to −5 and add to −4, so the answer is 3(x − 5)(x + 1). If the leading coefficient is negative, factor out the negative sign too: −2x² + 8 = −2(x² − 4) = −2(x − 2)(x + 2).
Special factoring patterns
- Difference of squares: a² − b² = (a − b)(a + b). For example, x² − 9y² = (x − 3y)(x + 3y).
- Perfect square trinomial: a² ± 2ab + b² = (a ± b)². In 4x² − 12x + 9, a = 2x and b = 3, and 2ab = 12x matches the middle term, so it equals (2x − 3)².
- Sum of cubes: a³ + b³ = (a + b)(a² − ab + b²).
- Difference of cubes: a³ − b³ = (a − b)(a² + ab + b²). For example, 8x³ − 125 = (2x − 5)(4x² + 10x + 25).
A sum of two squares such as x² + 1 does not factor over the real numbers. When both patterns fit, as in x⁶ − 64, use the difference of squares first: (x³ − 8)(x³ + 8). The two cube formulas then finish it.
Factoring by grouping
With four terms, factor each pair separately and look for a common binomial. For x³ + 2x² + 3x + 6: x²(x + 2) + 3(x + 2) = (x² + 3)(x + 2). Grouping also works with two letters: xy + 2x + 3y + 6 = (x + 3)(y + 2). Three terms can also group into a perfect square: x² − 4x + 4 − y² = (x − 2)² − y².
Higher-degree polynomials and the Rational Root Theorem
Every rational root p/q of a polynomial with integer coefficients has p dividing the constant term and q dividing the leading coefficient. For x³ − 6x² + 11x − 6, the candidates are ±1, ±2, ±3 and ±6. Substituting x = 1 gives 0, so (x − 1) is a factor. Synthetic division leaves x² − 5x + 6 = (x − 2)(x − 3).
Some polynomials are quadratics in disguise. x⁴ − 5x² + 4 is a quadratic in u = x², so it factors as (x² − 1)(x² − 4) = (x − 1)(x + 1)(x − 2)(x + 2). A polynomial with no rational roots can still split into quadratic factors. For example, x⁴ + 4 = (x² − 2x + 2)(x² + 2x + 2), and the calculator searches for such factors in polynomials up to degree 16.
When a polynomial is prime
A quadratic ax² + bx + c with integer coefficients factors over the rational numbers only when its discriminant b² − 4ac is a perfect square. For x² + 2x + 5 the discriminant is 4 − 20 = −16, so the quadratic is prime. Its roots are the complex numbers −1 ± 2i.
Factoring whole numbers
To factor a number, divide by primes that go in evenly: 360 → 180 → 90 → 45 → 15 → 5, so 360 = 2³ × 3² × 5. To count the factors, add 1 to each exponent and multiply: (3 + 1)(2 + 1)(1 + 1) = 24 factors. For a factor tree, use the prime factorization calculator.
Common factoring mistakes
- Forgetting the GCF, which leaves an answer like (2x − 4)(x + 1) that can still be factored.
- Getting the signs wrong: if c is positive, both numbers have the same sign as b; if c is negative, the numbers have opposite signs.
- Writing x² + 9 as (x + 3)². That square expands to x² + 6x + 9, not x² + 9.
- Using (a − b)² for a difference of squares. The correct form is (a − b)(a + b).
- Stopping too early, for example leaving x⁴ − 16 as (x² − 4)(x² + 4) when x² − 4 still factors.
How to check a factored answer
Multiply the factors back together. You should get the original polynomial. You can also substitute a value: with x = 1, 2x² + 7x + 3 = 12 and (2 · 1 + 1)(1 + 3) = 12. When your working differs from the calculator’s steps, compare them line by line to find the first step that differs.
Factoring practice quiz: 10 questions
Test yourself with 10 multiple-choice questions. Choose an answer and press Submit to see if you are right and why. Your score appears at the end.
Find two numbers that multiply to 6 and add to 5: 2 and 3. The pair 1 and 6 multiplies to 6 but adds to 7.
x² − 49 is a difference of squares, x² − 7², so it equals (x − 7)(x + 7). (x − 7)² would expand to x² − 14x + 49.
6 is the largest number dividing 12 and 18, and x² is the lowest power of x in both terms.
ac = 6, and 1 and 6 add to 7. Split: 2x² + x + 6x + 3 = x(2x + 1) + 3(2x + 1) = (2x + 1)(x + 3). The choice (2x + 3)(x + 1) expands to 2x² + 5x + 3.
Take out the GCF 3 to get 3(x² − 4), then factor the difference of squares: 3(x − 2)(x + 2). Stopping at 3(x² − 4) is not complete.
Difference of cubes: a³ − b³ = (a − b)(a² + ab + b²) with a = x and b = 2.
x² − 10x + 25 = (x − 5)², since 2 · x · 5 = 10x. A perfect square trinomial needs a positive last term and a middle term of ±2ab.
Its discriminant 3² − 4(1)(5) = −11 is not a perfect square, so it is prime. The others are (x + 1)(x + 5), (x − 5)(x + 1) and (x − 2)(x − 3).
72 = 2 × 2 × 2 × 3 × 3 = 2³ × 3². The choices 2³ × 9 and 8 × 9 equal 72 but are not written with primes.
Quiz complete
Frequently asked questions
How do you factor trinomials?
For x² + bx + c, find two numbers that multiply to c and add to b, then write (x + m)(x + n). When a is not 1, use the ac-method: find two numbers that multiply to ac and add to b. Split the middle term with them, then factor by grouping.
How do you factor completely?
Take out the GCF, then factor what is left with a pattern, a trinomial method or grouping. Repeat on every factor. For example, 3x² − 12x − 15 = 3(x − 5)(x + 1), and x² − 9 = (x − 3)(x + 3).
How do you solve an equation by factoring?
Move every term to one side, factor, and set each factor equal to 0 (the zero product property). For x² − x − 6 = 0, (x − 3)(x + 2) = 0, so x = 3 or x = −2.
Can you do factoring on a calculator?
The TI-84 has no factor command. Graph y = x² + bx + c and use the zero feature to find each root r, then write the factor (x − r). Calculators with a computer algebra system, such as the TI-Nspire CAS, can factor directly.
How do you factor polynomials with four or more terms?
Try the GCF and grouping first. For degree 3 or higher, use the Rational Root Theorem to find a root r, divide by (x − r), and factor the quotient. For example, x³ + 2x² + 3x + 6 = (x² + 3)(x + 2).
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