Greatest Common Factor First
A.SSE.2
Read the standard text
A.SSE.2 Use the structure of an expression to identify ways to rewrite it. For example, to factor 3x(x − 5) + 2(x − 5), students should recognize that the "x − 5" is common to both expressions being added, so it simplifies to (3x + 2)(x − 5); or see x^4 − y^4 as (x^2)^2 − (y^2)^2, thus recognizing it as a difference of squares that can be factored as (x^2 − y^2)(x^2 + y^2).
Section 1 · LearnLearn the idea
In this lesson you will: Find the greatest common factor of the terms of a polynomial; Factor it out, and check by distributing back; Recognize a common binomial factor, not just a common number or variable.
Distributing, run backwards
You already know that 3x(2x+5)=6x^2+15x. Factoring out a greatest common factor is that sentence read right to left. Nothing about the expression changes — only the way it is written.
The greatest common factor of the terms is the largest number that divides every coefficient, times the lowest power of each variable that appears in every term. In 6x^2+15x the coefficients 6 and 15 share a 3, and both terms carry at least one x, so the GCF is 3x.
Why it comes first, every time
Pulling the common factor out shrinks the numbers you then have to work with. 3x^2-12x-36 looks unpleasant. Take out the 3 and you are left with x^2-4x-12, which is the friendliest kind of trinomial there is — and the one Lesson 15.2 shows you how to finish.
That is not a coincidence. Ohio's content limits say that once a greatest common factor has been removed, the quadratic left behind is the kind you have been taught to factor. Skipping the GCF step is how students end up staring at a problem that was never supposed to be hard.
The common factor can be a whole expression
In 2x(x+4)+5(x+4) the repeated piece is (x+4). It behaves exactly like a common 3 or a common x would: pull it out front and collect what is left.
That gives (x+4)(2x+5). This move looks like a curiosity now. In Lesson 15.3 it is the last step of every problem.
Common mistake — Taking out a common factor, but not the greatest one
Asked to factor 12x^2+18x, a student writes 2x(6x+9).
Everything about that line is true. The 2x really is common, the distribution really does check, and the work looks finished.
It is not finished. The leftover 6x+9 still has a 3 in both terms, so the greatest common factor was 6x, not 2x. The complete answer is 6x(2x+3).
A partly factored answer is marked wrong on an item that says factor completely, and it also fails to do the job — the point of factoring is to break the expression down as far as it goes.
How to catch it: Look at the expression left inside the parentheses and ask the original question again: do these coefficients share anything? 6 and 9 share a 3, so you are not done. Keep asking until the answer is no.
Section 2 · DoPractice this standard
These published practice sets are scored on the student site and cover A.SSE.2 directly.