What Factoring Actually Means
Before we talk about how to factor, let's talk about why. Students often learn all the factoring techniques without anyone explaining what factoring is actually for. So here's the point. When you factor a polynomial, you rewrite it as a product. And that product form is useful because of one key rule: if two things multiply together to give zero, at least one of them has to be zero.
Factoring: x² + 3x − 4 = (x + 4)(x − 1)
That principle is called the Zero Product Property. It's why factoring matters so much when you're solving polynomial equations. Factor the polynomial, set each factor equal to zero, and one messy equation turns into two simple linear ones. That's the whole payoff.
Checking your work is always the same: expand your factored answer using the distributive property or FOIL. If you recover the original polynomial, you're done. If not, something went wrong, and it's much better to catch it now than to carry the mistake further.
Always Factor Out the GCF First
Before you try any other technique, scan every term in the polynomial for a shared factor. If one exists, pull it out. This is called factoring out the Greatest Common Factor (GCF), and it's honestly the most important habit in all of factoring. Do it first, every time. It makes everything that follows a lot simpler.
To find the GCF, look at two things: the GCF of the numerical coefficients, and the lowest power of any variable that appears in every term. Multiply those together and you have your GCF.
GCF of 10, 15, 5 = 5
Lowest power of x common to all terms = x²
GCF = 5x²
10x⁴ − 15x³ + 5x² = 5x²(2x² − 3x + 1)
Once you've pulled out the GCF, look at what's left inside the bracket. If it factors further using another technique, apply that technique now. A factorable expression still sitting inside the bracket means the problem isn't fully done.
GCF = 3ab
6a²b − 9ab² = 3ab(2a − 3b)
Even when the GCF is just a plain number with no variables, it's still worth pulling out. Smaller coefficients inside the bracket make every remaining step much easier to manage.
Factor by Grouping for Four-Term Expressions
When a polynomial has four terms and no obvious overall GCF, grouping is the go-to method. Split the four terms into two pairs, factor the GCF out of each pair separately, and then look for a common binomial across both groups. If you find one, factor it out once more and you're done.
Group into pairs:
(2x³ + 6x²) + (5x + 15)
Factor GCF from each pair:
2x²(x + 3) + 5(x + 3)
Both groups share (x + 3) - factor it out:
(x + 3)(2x² + 5)
The whole technique depends on getting a matching binomial from both pairs. If they match, grouping works cleanly. If they don't, try reordering the four terms before concluding it won't work. Sometimes a different pairing produces the match you need.
(3xy − 12x) + (2y − 8)
3x(y − 4) + 2(y − 4)
(y − 4)(3x + 2)
Factoring Simple Trinomials
A trinomial of the form x² + bx + c can often be factored into two binomials (x + p)(x + q). When you expand (x + p)(x + q), you get x² + (p + q)x + pq. So you're looking for two numbers that multiply to give c and add to give b. Find those two numbers and you're essentially done.
Need: two numbers that multiply to 12 and add to 8
Try factor pairs of 12: (1,12), (2,6), (3,4)
Check: 2 + 6 = 8 ✓ and 2 × 6 = 12 ✓
x² + 8x + 12 = (x + 2)(x + 6)
Signs are where students often trip up. When c is positive, p and q share the same sign as b, both positive if b is positive, both negative if b is negative. When c is negative, p and q have opposite signs, and the one with the larger absolute value takes b's sign.
Need: multiply to −18, add to −3
Try: −6 × 3 = −18 and −6 + 3 = −3 ✓
= (x − 6)(x + 3)
When the leading coefficient isn't 1, the process takes a bit more work. But the core idea is the same: find two numbers whose product and sum match values derived from the coefficients, use those numbers to split the middle term, then group.
The Difference of Squares Shortcut
When you see an expression of the form a² − b², two perfect squares with a minus sign between them, you can factor it instantly. The pattern is always a² − b² = (a + b)(a − b). No trial and error, no guessing. Just spot the pattern and apply it.
x² − 36 = (x + 6)(x − 6)
9x² − 4 = (3x + 2)(3x − 2)
25a² − 49b² = (5a + 7b)(5a − 7b)
The key skill is recognizing perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, and any variable raised to an even power, since x⁴ = (x²)², x⁶ = (x³)², and so on.
One important caution: the sum of squares (a² + b²) does not factor over real numbers. A frequent mistake is trying to factor x² + 25 as (x + 5)(x + 5), but that multiplies out to x² + 10x + 25, which is a completely different expression. Only the difference of squares has this shortcut. If the sign between the two squares is a plus, the binomial is already in its simplest form.