Introduction to Factoring: GCF
Factor out the greatest common factor from polynomial expressions. Recognize factoring as the reverse of distribution and verify answers by expanding. Apply to simplifying expressions and early equation solving.
A lesson of Algebra 1, in Polynomials, about 30 minutes.
By the end of this lesson, you will be able to find the greatest common factor of a polynomial's terms, factor it out, check your answer by multiplying back, and recognize why this is always the first thing to try.
Distribution, Backwards
You have spent the last three lessons multiplying. This lesson runs one of those multiplications in reverse.
$$3x(x + 4) = 3x^2 + 12x \qquad \text{multiplying}$$
$$3x^2 + 12x = 3x(x + 4) \qquad \text{factoring}$$
Same true statement, read the other way. Factoring means writing an expression as a product, and the reason it is harder than multiplying is that you have to find the factor rather than being handed it.
The check is free, and you should always do it. Multiply your answer back out. If it does not return exactly what you started with, the factoring is wrong, and you will know before anyone tells you.
The Greatest Common Factor
The GCF of a set of terms is the largest thing that divides into all of them. Build it in two independent pieces.
$$12x^3 + 18x^2$$
The number. The largest number dividing both $12$ and $18$ is $6$.
The variables. Take each variable to the lowest power that appears in any term. $x^3$ and $x^2$ both contain $x^2$, so $x^2$ it is.
$$\text{GCF} = 6x^2$$
Lowest power, not highest, and this is the one place people go wrong. The GCF must divide into every term, and $x^3$ does not divide into $x^2$. A shared factor can only be as big as the smallest supply of it.
$$12x^3 + 18x^2 = 6x^2(2x + 3)$$
Divide each term by the GCF to get what goes inside the bracket: $12x^3 \div 6x^2 = 2x$ and $18x^2 \div 6x^2 = 3$.
Example 1: The Whole Method
Here is every step on one expression, worked all the way through. $$20x^4 - 15x^2$$
- What is the largest number dividing both 20 and 15? 5
- What is the lowest power of x appearing in the two terms? x^2
- So what is the GCF? 5x^2
- Divide 20x^4 by the GCF 4x^2
- Write the factored form 5x^2(4x^2 - 3)
Sort It: What Is the GCF
The fifth one matters. Not everything has a GCF worth pulling out, and when the GCF is $1$ you move on to another method rather than forcing this one.
Worked Example 1: Three Terms
$$18x^3 + 12x^2 - 6x$$
Worked Example 2: A Negative Leading Term
$$-8x^3 - 12x^2$$
Why This Is Always First
Two reasons, and the second is the one that makes it a rule rather than a habit.
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