SOLUTION: Can someone walk me through the steps for the following problem? Thank you! 10a^2+25a-15 Factor completely. If a polynomial is prime, state this.

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Question 674303: Can someone walk me through the steps for the following problem? Thank you!
10a^2+25a-15
Factor completely. If a polynomial is prime, state this.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

10a%5E2%2B25a-15 Start with the given expression.


5%282a%5E2%2B5a-3%29 Factor out the GCF 5.


Now let's try to factor the inner expression 2a%5E2%2B5a-3


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Looking at the expression 2a%5E2%2B5a-3, we can see that the first coefficient is 2, the second coefficient is 5, and the last term is -3.


Now multiply the first coefficient 2 by the last term -3 to get %282%29%28-3%29=-6.


Now the question is: what two whole numbers multiply to -6 (the previous product) and add to the second coefficient 5?


To find these two numbers, we need to list all of the factors of -6 (the previous product).


Factors of -6:
1,2,3,6
-1,-2,-3,-6


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


These factors pair up and multiply to -6.
1*(-6) = -6
2*(-3) = -6
(-1)*(6) = -6
(-2)*(3) = -6

Now let's add up each pair of factors to see if one pair adds to the middle coefficient 5:


First NumberSecond NumberSum
1-61+(-6)=-5
2-32+(-3)=-1
-16-1+6=5
-23-2+3=1



From the table, we can see that the two numbers -1 and 6 add to 5 (the middle coefficient).


So the two numbers -1 and 6 both multiply to -6 and add to 5


Now replace the middle term 5a with -a%2B6a. Remember, -1 and 6 add to 5. So this shows us that -a%2B6a=5a.


2a%5E2%2Bhighlight%28-a%2B6a%29-3 Replace the second term 5a with -a%2B6a.


%282a%5E2-a%29%2B%286a-3%29 Group the terms into two pairs.


a%282a-1%29%2B%286a-3%29 Factor out the GCF a from the first group.


a%282a-1%29%2B3%282a-1%29 Factor out 3 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%2B3%29%282a-1%29 Combine like terms. Or factor out the common term 2a-1


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So 5%282a%5E2%2B5a-3%29 then factors further to 5%28a%2B3%29%282a-1%29


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Answer:


So 10a%5E2%2B25a-15 completely factors to 5%28a%2B3%29%282a-1%29.


In other words, 10a%5E2%2B25a-15=5%28a%2B3%29%282a-1%29.


Note: you can check the answer by expanding 5%28a%2B3%29%282a-1%29 to get 10a%5E2%2B25a-15 or by graphing the original expression and the answer (the two graphs should be identical).