What does it mean to factor an expression?
Factoring means rewriting a sum as a product. It is the reverse of expanding: (2x + 5)(3x − 2) multiplies out to 6x² + 11x − 10, so factoring 6x² + 11x − 10 takes you back to (2x + 5)(3x − 2).
A product is easier to work with. If you need to solve 6x² + 11x − 10 = 0, the factored form tells you right away that 2x + 5 = 0 or 3x − 2 = 0. Factors also make fractions simpler, because a shared factor on the top and bottom cancels.
What order does the factoring calculator follow?
The factoring calculator takes out the greatest common factor first, then counts the terms to pick a method. Follow the same order by hand. That one habit prevents most wrong answers.
- Pull out the GCF of every term, including a negative sign if the leading coefficient is negative.
- Two terms: check for a difference of squares, a² − b² = (a + b)(a − b), or a sum or difference of cubes.
- Three terms: check for a perfect square trinomial, then use the ac method.
- Four terms: try factoring by grouping.
- Degree 3 or higher with no pattern: use the rational root theorem to find a root, then divide it out.
- Look at every factor again and keep factoring until none can be split.
For example, 3x² − 12 has a GCF of 3. Taking it out leaves 3(x² − 4), and x² − 4 is a difference of squares, so the full answer is 3(x + 2)(x − 2).
How do you factor trinomials with the ac method?
Multiply a by c, find two numbers that multiply to ac and add to b, split the middle term with those numbers, and factor by grouping. This works for any trinomial ax² + bx + c that factors over the integers, and it is what a factoring trinomials calculator does behind the scenes.
Here is 6x² + 11x − 10 worked through. The product a·c is 6 × (−10) = −60, and the pair that multiplies to −60 and adds to 11 is −4 and 15.
Multiply the answer back out to check it. (2x + 5)(3x − 2) gives 6x² − 4x + 15x − 10, which is 6x² + 11x − 10. The calculator runs this same check on every answer it gives.
Which special factoring patterns should you know?
The two you will see most are the difference of squares and the perfect square trinomial. Spotting them saves you from running the full ac method.
A difference of squares has two perfect squares with a minus sign between them. In 4x² − 9, the terms are (2x)² and 3², so it factors as (2x + 3)(2x − 3). A sum of squares like x² + 4 does not factor over the real numbers.
A perfect square trinomial has a first and last term that are squares, and a middle term equal to twice their product. In 4x² + 12x + 9, the middle term is 2 × 2x × 3 = 12x, so the answer is (2x + 3)².
| Pattern | Formula | Example |
|---|---|---|
| Difference of squares | a² − b² = (a + b)(a − b) | 4x² − 9 = (2x + 3)(2x − 3) |
| Perfect square (plus) | a² + 2ab + b² = (a + b)² | x² + 6x + 9 = (x + 3)² |
| Difference of cubes | a³ − b³ = (a − b)(a² + ab + b²) | x³ − 27 = (x − 3)(x² + 3x + 9) |
How do you factor polynomials of higher degree?
Start with the GCF, then look for a pattern or a rational root in what remains. Higher degree problems often turn into a quadratic once the common factor is gone.
Take −5h⁴ + 30h³ − 40h². Every term has h² and a factor of 5, and the leading term is negative, so take out −5h². That leaves −5h²(h² − 6h + 8). Two numbers that multiply to 8 and add to −6 are −2 and −4, giving −5h²(h − 2)(h − 4).
With x⁴ − 1, the difference of squares works twice in a row. First x⁴ − 1 = (x² + 1)(x² − 1), then x² − 1 = (x + 1)(x − 1). The final answer is (x + 1)(x − 1)(x² + 1). The calculator reaches the same result by finding the roots 1 and −1 with the rational root theorem.
What does irreducible over the rationals mean?
It means the polynomial cannot be written as a product of smaller polynomials with whole number or fraction coefficients. The calculator shows this label when nothing more can be pulled out.
For example, x² + x + 1 is irreducible over the rationals because no two integers multiply to 1 and add to 1. Its discriminant is 1 − 4 = −3, which is not a perfect square. A quadratic only factors over the rationals when b² − 4ac is a perfect square.
Irreducible over the rationals does not always mean irreducible everywhere. The polynomial x² − 5 has the roots √5 and −√5, so it splits as (x − √5)(x + √5) if square roots are allowed. Most algebra classes stop at rational coefficients, which is why the calculator leaves x² − 5 as it is.
What are the most common factoring mistakes?
Forgetting the GCF is the big one. If you factor 2x² + 8x + 8 as (2x + 4)(x + 2), you are not finished, because 2x + 4 still has a 2 in it. The complete answer is 2(x + 2)².
Sign errors come next. When c is negative, the two numbers in the ac method have opposite signs. When c is positive and b is negative, both numbers are negative. Students also stop early on x⁴ − 1, writing (x² + 1)(x² − 1) and missing that x² − 1 splits again.