SOLUTION: Factor Completely. I have been struggling with trinomials and all nomial but could I get some insight on this problem. Our books are written horribly for us normal math folks. I

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: Factor Completely. I have been struggling with trinomials and all nomial but could I get some insight on this problem. Our books are written horribly for us normal math folks. I      Log On


   

Question 370538: Factor Completely. I have been struggling with trinomials and all nomial but could I get some insight on this problem. Our books are written horribly for us normal math folks. I could use some common sense explanation
Factor Completely:
a^2b + 2ab - 15b
This site has been a savior because of the excellent explaination written for the common person.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

a%5E2b%2B2ab-15b Start with the given expression.


b%28a%5E2%2B2a-15%29 Factor out the GCF b.


Now let's try to factor the inner expression a%5E2%2B2a-15


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Looking at the expression a%5E2%2B2a-15, we can see that the first coefficient is 1, the second coefficient is 2, and the last term is -15.


Now multiply the first coefficient 1 by the last term -15 to get %281%29%28-15%29=-15.


Now the question is: what two whole numbers multiply to -15 (the previous product) and add to the second coefficient 2?


To find these two numbers, we need to list all of the factors of -15 (the previous product).


Factors of -15:
1,3,5,15
-1,-3,-5,-15


Note: list the negative of each factor. This will allow us to find all possible combinations.


These factors pair up and multiply to -15.
1*(-15) = -15
3*(-5) = -15
(-1)*(15) = -15
(-3)*(5) = -15

Now let's add up each pair of factors to see if one pair adds to the middle coefficient 2:


First NumberSecond NumberSum
1-151+(-15)=-14
3-53+(-5)=-2
-115-1+15=14
-35-3+5=2



From the table, we can see that the two numbers -3 and 5 add to 2 (the middle coefficient).


So the two numbers -3 and 5 both multiply to -15 and add to 2


Now replace the middle term 2a with -3a%2B5a. Remember, -3 and 5 add to 2. So this shows us that -3a%2B5a=2a.


a%5E2%2Bhighlight%28-3a%2B5a%29-15 Replace the second term 2a with -3a%2B5a.


%28a%5E2-3a%29%2B%285a-15%29 Group the terms into two pairs.


a%28a-3%29%2B%285a-15%29 Factor out the GCF a from the first group.


a%28a-3%29%2B5%28a-3%29 Factor out 5 from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.


%28a%2B5%29%28a-3%29 Combine like terms. Or factor out the common term a-3


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So b%28a%5E2%2B2a-15%29 then factors further to b%28a%2B5%29%28a-3%29


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Answer:


So a%5E2b%2B2ab-15b completely factors to b%28a%2B5%29%28a-3%29.


In other words, a%5E2b%2B2ab-15b=b%28a%2B5%29%28a-3%29.


Note: you can check the answer by expanding b%28a%2B5%29%28a-3%29 to get a%5E2b%2B2ab-15b or by graphing the original expression and the answer (the two graphs should be identical).


If you need more help, email me at jim_thompson5910@hotmail.com

Also, feel free to check out my tutoring website

Jim