Factoring Completely
Factor polynomials completely by first extracting the GCF, then applying the appropriate factoring technique to the remaining factor. Recognize when a polynomial requires multiple factoring steps.
A lesson of Algebra 1, in Factoring Polynomials, about 30 minutes.
By the end of this lesson, you will be able to run the factoring decision in a fixed order, keep going until nothing is left to factor, factor by grouping when there are four terms, and know when to stop.
One Order, Every Time
You now have four methods. The skill left is choosing between them, and it is mechanical if you always ask the same questions in the same order.
- GCF. Always. Pull out whatever every term shares
- Count the terms in what is left
- Two terms: difference of squares?
- Three terms: perfect square, then $x^2 + bx + c$, then AC
- Four terms: group in pairs
- Check every factor you produced. Can any of them factor again?
Step 3 is the one people skip, and it is what "completely" means.
$$2x^3 - 8x = 2x(x^2 - 4) = 2x(x + 2)(x - 2)$$
Stopping at $2x(x^2-4)$ is a correct GCF and an incomplete answer, because $x^2 - 4$ is a difference of squares and was sitting there in plain view.
Factored Completely Means Nothing Is Left
$$\text{a product of factors, none of which factors further}$$
The practical test is to look at each bracket in turn and try the decision again.
- Two terms: is it a difference of squares? Are both squares?
- Three terms: do two numbers multiply to $ac$ and add to $b$?
- Any linear factor like $(x+3)$ is finished. Degree one never factors
A useful sanity check: the degrees of your factors should add up to the degree you started with. $2x^3 - 8x$ has degree $3$, and $2x \cdot (x+2)(x-2)$ has degrees $1 + 1 + 1 = 3$. If the total falls short, you left something factored too little.
Example 1: The Whole Decision
Here is every step on one expression, worked all the way through. $$3x^3 - 27x$$
- What is the GCF of the two terms? 3x
- Factor it out. What is inside the bracket? x^2 - 9
- How many terms are in the bracket? 2
- Two terms, both squares, subtracted. Factor the bracket (x + 3)(x - 3)
- Write the completely factored form 3x(x + 3)(x - 3)
Factoring by Grouping: Four Terms
Four terms with no common factor across all of them usually means grouping.
$$x^3 + 3x^2 + 2x + 6$$
Split into two pairs and factor each pair separately.
$$x^2(x + 3) + 2(x + 3)$$
$$(x + 3)(x^2 + 2)$$
The two brackets must match, and that is both the mechanism and the check. You already used this inside the AC method; here it is the whole method.
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