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Showing posts with label factor. Show all posts
Showing posts with label factor. Show all posts

## Sunday, November 4, 2012

### General Guidelines for Factoring Polynomials

Now that we have learned techniques for factoring 4, 3, and 2-term polynomials, we are ready to practice by mixing up the problems. The challenge is to first identify the type of factoring problem then decide which method to apply. The basic guidelines for factoring follow:

1. Look to factor out any GCF.
2. Four-Term Polynomials - Factor by grouping.
3. Trinomials - Factor using the "guess and check" method.
4. Binomials - Use the the special products in this order:
Sum and Difference of Squares

Sum and Difference of Cubes

* If a binomial is both a difference of squares and cubes, then to obtain a more complete factorization, factor it as a difference of squares first.
* Not all polynomials factor.  In this case, beginning algebra students may write, "does not factor - DNF."

Factor.
Tip: Make some note cards to aid in helping memorize the formulas for the special products. Look for factors to factor further - sometimes factoring once is not enough.

Factor.
Take some time to understand the difference between the last two solved problems. Notice that x^6 - y^6  is both a difference of squares and a difference of cubes at the same time. Here we chose to apply the difference of squares formula first. On the other hand, for x^6 + y^6 we chose to apply the sum of cubes formula first because it does not factor as a sum of squares.

Factor.
Video Examples on YouTube: Factor the following polynomials.

### Factoring Special Binomials

Here we will learn how to factor special binomials. We have seen some of the formulas in a previous section, however, we will be using them in a different manner here. These formulas will be used as a template for factoring.
The idea is to identify a and b and then plug in to the formulas above.

Factor:
Step 1: Identify the special binomial.
Step 2: Identify the a and the b in the formula.
Step 3: Substitute into the appropriate formula.
Step 4: Check by multiplying.
The same steps can be used for the sum and difference of cubes formula.
Review the perfect cubes up to 10. Many times the coefficients will give a clue as to what special binomial formula is to be used.

Factor.
At this point we will look a bit more closely at the process. First identify the binomial that is to be factored.
In this case, we have a sum of cubes were a = m and b = 2n. Next identify the appropriate formula and substitute into it.
After identifying the special binomial and determining the a and the b, it is just a matter of substituting into the appropriate formula. Much of this can be done mentally, so it is sufficient to present your solution without the above steps.
Rest assured that with much practice you will be able to jump straight to the answer too. The first step toward this ability, of course, is to memorize the formulas.
As we have seen before, we will often run into polynomials with a GCF. It is important with special binomials to factor out the GCF first.

Factor:
As it stands, this binomial is not a difference of squares. Factor out the GCF and see that the remaining factor is a difference of squares.
The GCF is still part of the answer.

Factor.
Factoring binomials is a bit more complicated when larger exponents are involved. It is difficult to recognize that x^6, for example, is a perfect cube. We can think of x^6 = (x^2)^3  or the cube of x squared. Also, recall the rule of exponents
Factor
Sum of cubes.

It is not always necessary to show all the steps shown above. Ask your instructor what he or she wants to see in the way of steps when presenting your solutions in this case.

Factor.
Look for factors that factor further. In other words, look to continue factoring until all factors are completely factored. Also, the trinomials that we obtain when using the sum and difference of cubes will not factor.
If we are confronted with a polynomial that is both a difference of squares and a difference of cubes we must factor it as a difference of squares first. Doing this will ensure that we obtain a complete factorization.

Video Examples on YouTube: Factor the special binomials below.