Always start with the GCF
Before anything else, check whether every term shares a factor. In 3x^2 + 21x + 36, all three coefficients divide by 3, so write it as 3(x^2 + 7x + 12). The trinomial left inside is much easier to work with.
Skipping this step is why people end up with factors like (3x + 9)(x + 4). That's correct in a sense, but it isn't fully factored, because 3x + 9 still contains a 3.
When a = 1: find two numbers
For x^2 + bx + c, you need two numbers that multiply to c and add to b. For x^2 + 7x + 12, try factor pairs of 12: 1 and 12 add to 13, 2 and 6 add to 8, 3 and 4 add to 7. That's it. The answer is (x + 3)(x + 4), so 3x^2 + 21x + 36 = 3(x + 3)(x + 4).
Signs help you search faster. If c is positive, both numbers have the same sign as b. If c is negative, the numbers have opposite signs, and the bigger one takes the sign of b. For x^2 - 2x - 15, you want numbers with opposite signs that multiply to -15 and add to -2. That's -5 and 3, so the answer is (x - 5)(x + 3).
- c positive, b positive: both numbers positive
- c positive, b negative: both numbers negative
- c negative: one positive, one negative
When a is bigger than 1: the AC method
Multiply a by c. Find two numbers that multiply to ac and add to b. Use them to split the middle term into two pieces, then factor by grouping.
Take 6x^2 - 7x - 3. Here ac = -18. You need two numbers that multiply to -18 and add to -7: that's -9 and 2. Rewrite the middle: 6x^2 - 9x + 2x - 3. Group the first two terms and the last two: 3x(2x - 3) + 1(2x - 3). Both groups share 2x - 3, so the answer is (3x + 1)(2x - 3).
Spot the special patterns
A few trinomials and binomials let you skip the search. If the first and last terms are perfect squares and the middle term is twice their roots multiplied, it's a perfect square: 4x^2 - 4x + 1 = (2x - 1)^2. If there's no middle term and you're subtracting two squares, it's a difference of squares: x^2 - 9 = (x - 3)(x + 3).
A sum of squares, like x^2 + 9, doesn't factor over the real numbers. That catches people out on tests.
How to tell when it won't factor
Sometimes no pair of numbers works, and that's not your fault. Compute b^2 - 4ac. If it isn't a perfect square, the trinomial doesn't factor with whole numbers. For x^2 + x + 1 it's 1 - 4 = -3, so stop searching and use the quadratic formula if you need the roots.
This test saves a lot of time on big coefficients. Our factoring calculator runs it for you and multiplies the answer back out, which is handy for checking homework.
Check by multiplying back
Every factoring answer can be checked in a few seconds. Multiply the binomials using first, outer, inner, last, and compare with the original. For (3x + 1)(2x - 3): 6x^2 - 9x + 2x - 3 = 6x^2 - 7x - 3. It matches.
If the middle term comes out with the wrong sign, swap the signs inside the binomials. If it's the right size but wrong, try a different factor pair.
Frequently asked questions
What is the AC method?
It's a way to factor ax^2 + bx + c when a isn't 1. Multiply a and c, find two numbers that multiply to ac and add to b, split the middle term, then factor by grouping.
How do I factor x^2 + 7x + 12?
Find two numbers that multiply to 12 and add to 7. Those are 3 and 4, so it factors as (x + 3)(x + 4).
Can every trinomial be factored?
No. If b^2 - 4ac isn't a perfect square, the trinomial won't factor using whole numbers. x^2 + x + 1 is an example.
Why do I need to take out the GCF first?
Without it your answer isn't fully factored, and the remaining trinomial is harder to work with. Pulling out a 3 from 3x^2 + 21x + 36 leaves the simpler x^2 + 7x + 12.
Does x^2 + 9 factor?
Not over the real numbers. A sum of squares has no real factors. Only a difference of squares, like x^2 - 9, splits into two binomials.
References
These sources support the method or guidance used for Factoring Calculator. Verify time-sensitive rules at the source.
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