- The Method of Factorization
Summary: This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. Factoring is an essential skill for success in algebra and higher level mathematics courses. Therefore, we have taken great care in developing the student's understanding of the factorization process. The technique is consistently illustrated by displaying an empty set of parentheses and describing the thought process used to discover the terms that are to be placed inside the parentheses. The factoring scheme for special products is presented with both verbal and symbolic descriptions, since not all students can interpret symbolic descriptions alone. Two techniques, the standard "trial and error" method, and the "collect and discard" method (a method similar to the "ac" method), are presented for factoring trinomials with leading coefficients different from 1. Objectives of this module: be able to factor trinomials with leading coefficient other than 1.
In the last section we saw that we could easily factor trinomials of the form
Factor
The third term of the trinomial is
(a) product is
(b) sum is
The required numbers are
The problem of factoring the polynomial
Consider the product
Examining the trinomial
6 | |
Look for the combination that when multiplied and then added yields the middle term.
The proper combination we're looking for is
Factor
Factor the first and last terms.
Thus,
Factor
Find the factors of the first and last terms.
Thus, the
Factor
Before we start finding the factors of the first and last terms, notice that the constant term is
Factor the first and last terms.
After a few trials we see that
Factor
We see that each term is even, so we can factor out 2.
Notice that the constant term is positive. Thus, we know that the factors of 3 that we are looking for must have the same sign. Since the sign of the middle term is negative, both factors must be negative.
Factor the first and last terms.
There are not many combinations to try, and we find that
If we had not factored the 2 out first, we would have gotten the factorization
The factorization is not complete since one of the factors may be factored further.
The results are the same, but it is much easier to factor a polynomial after all common factors have been factored out first.
Factor
There are no common factors. We see that the constant term is negative. Thus, the factors of
Factor the first and last terms.
After a few trials, we see that
Factor
We see that the constant term is positive and that the middle term is preceded by a minus sign.
Hence, the factors of
Factor the first and last terms.
After a few trials, we see that
Factor the following, if possible.
not factorable
As you get more practice factoring these types of polynomials you become faster at picking the proper combinations. It takes a lot of practice!
There is a shortcut that may help in picking the proper combinations. This process does not always work, but it seems to hold true in many cases. After you have factored the first and last terms and are beginning to look for the proper combinations, start with the intermediate factors and not the extreme ones.
Factor
Factor the first and last terms.
12 | |
Rather than starting with the
Factor
Factor
Consider the polynomial
Now, compute
Find the factors of
But we have included too much. We must eliminate the surplus. Factor each parentheses.
Discard the factors that multiply to
Factor
Identify
Compute
Find the factors of
We have collected too much. Factor each set of parentheses and eliminate the surplus.
Discard the factors that multiply to
Factor
Identify
Compute
Find the factors of
We have collected too much. Factor each set of parentheses and eliminate the surplus.
Discard the factors that multiply to
Factor
Identify
Compute
Find the factors of
We have collected too much. Factor each set of parentheses and eliminate the surplus.
Discard the factors that multiply to
Factor
Factor
Factor
Factor
Factor the following problems, if possible.
not factorable
not factorable
For the following problems, the given trinomial occurs when solving the corresponding applied problem. Factor each trinomial. You do not need to solve the problem.
It takes 5 hours to paddle a boat 12 miles downstream and then back. The current flows at the rate of 1 mile per hour. At what rate was the boat paddled?
The length of a rectangle is 5 inches more than the width of the rectangle. If the area of the rectangle is 84 square inches, what are the length and width of the rectangle?
A square measures 12 inches on each side. Another square is to be drawn around this square in such a way that the total area is 289 square inches. What is the distance from the edge of the smaller square to the edge of the larger square? (The two squares have the same center.)
A woman wishes to construct a rectangular box that is open at the top. She wishes it to be 4 inches high and have a rectangular base whose length is three times the width. The material used for the base costs $2 per square inch, and the material used for the sides costs $1.50 per square inch. The woman will spend exactly $120 for materials. Find the dimension of the box (length of the base, width of the base, and height).
For the following problems, factor the trinomials if possible.
not factorable
((Reference)) Simplify
((Reference)) Find the product.
((Reference)) Find the product.
((Reference)) Solve the equation
((Reference)) Factor
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