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2.01 Factoring by Grouping Intermediate Algebra: One Step at a Time. Pages 131 - 138: 5, 11, 16, 33, 35, 45, 48.Dr. Robert J. Rapalje Seminole State College of Florida Sanford, FL 32773
To see Section 2.01, with explanations, examples, exercises, and answers, click here!
See also : Basic Algebra: Factoring by Grouping
While there are many different types of factoring by grouping, a good place to start, especially if there are four terms in the problem, is to try grouping the first two terms and the last two terms. When you group the first two and the last two terms together, you MUST get a common factor! If you don’t get a common factor—that is, if the second binomial factor does not match the first binomial factor EXACTLY, then you may have made an error. INTRODUCTORY PROBLEM .
SOLUTION: Notice that there are NO common factors, it is NOT a trinomial, and it is NOT the difference of squares or sum/difference of cubes! The only thing left to do is to try grouping the first two terms and the last two terms, and hope you get a common factor.
Notice that
you DO have a common factor
5.
SOLUTION: Notice that there are NO common factors, it is NOT a trinomial, and it is NOT the difference of two squares! The only thing left to do is to try a grouping of the first two terms and the last two terms, and hope you get a common factor.
Notice that
you DO have a common factor
11.
SOLUTION: Notice that there are NO common factors, it is NOT a trinomial, and it is NOT the difference of two squares! Group the first two terms and the last two terms, and make sure that you get a common factor!
Notice that
you DO have a common factor
Now, you have a difference of two squares, which factors again!
16.
SOLUTION: Notice that there are NO common factors, it is NOT a trinomial, and it is NOT the difference of two squares! Group the first two terms and the last two terms, and make sure that you get a common factor!
Notice that
you DO have a common factor
Now, you have a difference of two squares, which factors again!
33.
The first step is to recognize that this is a trinomial. Can you see that it is in three parts?
The
FIRST times
FIRST must be The
FIRST times
FIRST must be
Next, find two numbers whose product is 60 and whose sum is 16. That would be 10 and 6:
This can be “cleaned up” to make it look like the product of two regular trinomials, and as “chance” would have it, these trinomials each can be factored!!
 Final Answer!!
35.
The first step is to recognize that this is a difference of two squares! The
FIRST times
FIRST must be The
FIRST times
FIRST must be
This can be “cleaned up” to make it look like the product of two regular trinomials,
and as “chance” would have it, these trinomials each can be factored!!
45. x2 + 2xy + y2 + 7x + 7y + 10Group the first three terms, the next two, and keep the last term separate. This begins to look like a trinomial!
Find two numbers whose product is 10 and whose sum is 7.
This cleans up to give you this for the final answer:
48.
First notice that, because of the number of terms involved here, this must be a grouping problem. Did you notice that the first three terms look good together? It turns out that these first three terms form a perfect square trinomial. Then try grouping the next two terms together from which you can factor out a common factor of 3. The last term stays by itself. Putting this into grouping by color may help you see it better:
Rewrite it in this form
and recognize that this is a trinomial. Can you see that it is in three parts? The
FIRST times
FIRST must be The FIRST times FIRST
must be
Next, find two numbers whose product is 2 and whose sum is 3. That would be 2 and 1:
This can be “cleaned up” to make it look like the product of two regular trinomials,
From College Algebra:
The first step is to recognize that this is a trinomial. Can you see that it is in three parts? The
FIRST times
FIRST must be The
FIRST times
FIRST must be
Next, find two numbers whose product is 8 and whose sum is −9. That would be −8 and −1: Each of these factors represent the difference of
cubes, which can be factored using the formula:
These trinomials CANNOT be factored, so this is your final answer!!
From College Algebra: The first step is to recognize that this is a difference of two squares! The
FIRST times
FIRST must be The
FIRST times
FIRST must be
Each of these factors represent the difference or sum
of cubes, which can be factored using the formulas:
and
These trinomials CANNOT be factored, so this is your final answer!!
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