SOLUTION: Factor by grouping. (If the expression is nonfactorable enter PRIME.) 4a2 + 7ab - 2b2

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Question 729202: Factor by grouping. (If the expression is nonfactorable enter PRIME.)
4a2 + 7ab - 2b2
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

Looking at the expression 4a%5E2%2B7ab-2b%5E2, we can see that the first coefficient is 4, the second coefficient is 7, and the last coefficient is -2.


Now multiply the first coefficient 4 by the last coefficient -2 to get %284%29%28-2%29=-8.


Now the question is: what two whole numbers multiply to -8 (the previous product) and add to the second coefficient 7?


To find these two numbers, we need to list all of the factors of -8 (the previous product).


Factors of -8:
1,2,4,8
-1,-2,-4,-8


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


These factors pair up and multiply to -8.
1*(-8) = -8
2*(-4) = -8
(-1)*(8) = -8
(-2)*(4) = -8

Now let's add up each pair of factors to see if one pair adds to the middle coefficient 7:


First NumberSecond NumberSum
1-81+(-8)=-7
2-42+(-4)=-2
-18-1+8=7
-24-2+4=2



From the table, we can see that the two numbers -1 and 8 add to 7 (the middle coefficient).


So the two numbers -1 and 8 both multiply to -8 and add to 7


Now replace the middle term 7ab with -ab%2B8ab. Remember, -1 and 8 add to 7. So this shows us that -ab%2B8ab=7ab.


4a%5E2%2Bhighlight%28-ab%2B8ab%29-2b%5E2 Replace the second term 7ab with -ab%2B8ab.


%284a%5E2-ab%29%2B%288ab-2b%5E2%29 Group the terms into two pairs.


a%284a-b%29%2B%288ab-2b%5E2%29 Factor out the GCF a from the first group.


a%284a-b%29%2B2b%284a-b%29 Factor out 2b 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%2B2b%29%284a-b%29 Combine like terms. Or factor out the common term 4a-b


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Answer:


So 4a%5E2%2B7ab-2b%5E2 factors to %28a%2B2b%29%284a-b%29.


In other words, 4a%5E2%2B7ab-2b%5E2=%28a%2B2b%29%284a-b%29.


Note: you can check the answer by expanding %28a%2B2b%29%284a-b%29 to get 4a%5E2%2B7ab-2b%5E2 or by graphing the original expression and the answer (the two graphs should be identical).