Pull out what every term shares, then look for the pattern that splits a square into two.
A store sells boxes of pens. One box holds 6 pens, another holds 10, and a third holds 14. The store manager wants to repack everything into identical smaller boxes with no pens left over. She figures out that each smaller box must hold exactly 2 pens — because 2 divides into 6, 10, and 14 evenly. That reasoning — finding the largest number that divides cleanly into every term — is exactly what you do when you factor out a greatest common factor.
Factoring is the reverse of distributing. When you distribute, you multiply something into parentheses. When you factor, you pull something out. The greatest common factor, or GCF, of a polynomial is the largest expression that divides evenly into every single term.
Take the polynomial 6x2+10x. Every term is divisible by 2x. Pulling it out gives:
6x2+10x=2x(3x+5)
To check, distribute back: 2x⋅3x=6x2 and 2x⋅5=10x. You get the original polynomial, so the factoring is correct.
The process has two steps: find the GCF of the coefficients by asking what is the largest integer that divides into all of them, then find the GCF of the variable parts by taking the lowest power of each variable that appears in every term.
Factor 12x^3 + 8x^2 - 4x
gcd(12,8,4)=4
The largest integer that divides 12, 8, and 4 is 4.
variable GCF: x1
The lowest power of x across all three terms is x to the first — x³ has it, x² has it, and x has it.
4x(3x2+2x−1)
Divide each term by 4x: 12x³ ÷ 4x = 3x², 8x² ÷ 4x = 2x, 4x ÷ 4x = 1.
4x(3x2+2x−1)✓
Distribute back to verify: 4x·3x² = 12x³, 4x·2x = 8x², 4x·1 = 4x. Correct.
Factor 15x^4 - 10x^3 + 5x^2
gcd(15,10,5)=5
5 is the largest integer dividing all three coefficients.
variable GCF: x2
The lowest power of x is x² — it appears in x⁴, x³, and x².
Now for a different pattern entirely. Multiply out (x+4)(x−4) and watch what happens:
(x+4)(x−4)=x2−4x+4x−16=x2−16
The middle terms cancelled. That always happens when two binomials are identical except for the sign between them. The result is a difference of two perfect squares. Running it backwards: if you see a polynomial that is one perfect square minus another, it factors as two binomials — one with a plus, one with a minus.
a2−b2=(a+b)(a−b)
To use this, you need to recognize perfect squares. The number 25 is a perfect square because 25=52. The term x2 is a perfect square because x2=(x)2. The term 9x2 is a perfect square because 9x2=(3x)2.
Sometimes both techniques apply to the same polynomial — a GCF hides inside what looks like a difference of squares problem. Always factor out the GCF first. That step can reveal the pattern underneath.
Factor 2x^2 - 50
2(x2−25)
Both terms share a factor of 2. Pull it out first before looking for any other pattern.
x2=(x)2,25=(5)2
Now look at what is inside the parentheses. It is a difference of squares.
2(x+5)(x−5)
Factor the difference of squares inside. The 2 out front stays.