SOLUTION: Factor completely. If the polynomial cannot be factored, enter PRIME. Problem 7a2 + 63a - 70 I can't solve this, someone help.

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Question 319162: Factor completely. If the polynomial cannot be factored, enter PRIME.
Problem 7a2 + 63a - 70
I can't solve this, someone help.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

7a%5E2%2B63a-70 Start with the given expression.


7%28a%5E2%2B9a-10%29 Factor out the GCF 7.


Now let's try to factor the inner expression a%5E2%2B9a-10


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Looking at the expression a%5E2%2B9a-10, we can see that the first coefficient is 1, the second coefficient is 9, and the last term is -10.


Now multiply the first coefficient 1 by the last term -10 to get %281%29%28-10%29=-10.


Now the question is: what two whole numbers multiply to -10 (the previous product) and add to the second coefficient 9?


To find these two numbers, we need to list all of the factors of -10 (the previous product).


Factors of -10:
1,2,5,10
-1,-2,-5,-10


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


These factors pair up and multiply to -10.
1*(-10) = -10
2*(-5) = -10
(-1)*(10) = -10
(-2)*(5) = -10

Now let's add up each pair of factors to see if one pair adds to the middle coefficient 9:


First NumberSecond NumberSum
1-101+(-10)=-9
2-52+(-5)=-3
-110-1+10=9
-25-2+5=3



From the table, we can see that the two numbers -1 and 10 add to 9 (the middle coefficient).


So the two numbers -1 and 10 both multiply to -10 and add to 9


Now replace the middle term 9a with -a%2B10a. Remember, -1 and 10 add to 9. So this shows us that -a%2B10a=9a.


a%5E2%2Bhighlight%28-a%2B10a%29-10 Replace the second term 9a with -a%2B10a.


%28a%5E2-a%29%2B%2810a-10%29 Group the terms into two pairs.


a%28a-1%29%2B%2810a-10%29 Factor out the GCF a from the first group.


a%28a-1%29%2B10%28a-1%29 Factor out 10 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%2B10%29%28a-1%29 Combine like terms. Or factor out the common term a-1


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So 7%28a%5E2%2B9a-10%29 then factors further to 7%28a%2B10%29%28a-1%29


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Answer:


So 7a%5E2%2B63a-70 completely factors to 7%28a%2B10%29%28a-1%29.


In other words, 7a%5E2%2B63a-70=7%28a%2B10%29%28a-1%29.


Note: you can check the answer by expanding 7%28a%2B10%29%28a-1%29 to get 7a%5E2%2B63a-70 or by graphing the original expression and the answer (the two graphs should be identical).