SOLUTION: Factor the following into the product of two binomials by “reverse FOIL” method. Then, confirm your answer is correct by multiplying the two binomials: x2 + 8x + 7

Algebra ->  Distributive-associative-commutative-properties -> SOLUTION: Factor the following into the product of two binomials by “reverse FOIL” method. Then, confirm your answer is correct by multiplying the two binomials: x2 + 8x + 7       Log On


   

Question 103972: Factor the following into the product of two binomials by “reverse FOIL” method. Then, confirm your answer is correct by multiplying the two binomials:
x2 + 8x + 7
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)


Looking at the expression x%5E2%2B8x%2B7, we can see that the first coefficient is 1, the second coefficient is 8, and the last term is 7.



Now multiply the first coefficient 1 by the last term 7 to get %281%29%287%29=7.



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



To find these two numbers, we need to list all of the factors of 7 (the previous product).



Factors of 7:

1,7

-1,-7



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



These factors pair up and multiply to 7.

1*7 = 7
(-1)*(-7) = 7


Now let's add up each pair of factors to see if one pair adds to the middle coefficient 8:



First NumberSecond NumberSum
171+7=8
-1-7-1+(-7)=-8




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



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



Now replace the middle term 8x with x%2B7x. Remember, 1 and 7 add to 8. So this shows us that x%2B7x=8x.



x%5E2%2Bhighlight%28x%2B7x%29%2B7 Replace the second term 8x with x%2B7x.



%28x%5E2%2Bx%29%2B%287x%2B7%29 Group the terms into two pairs.



x%28x%2B1%29%2B%287x%2B7%29 Factor out the GCF x from the first group.



x%28x%2B1%29%2B7%28x%2B1%29 Factor out 7 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.



%28x%2B7%29%28x%2B1%29 Combine like terms. Or factor out the common term x%2B1



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Answer:



So x%5E2%2B8%2Ax%2B7 factors to %28x%2B7%29%28x%2B1%29.



In other words, x%5E2%2B8%2Ax%2B7=%28x%2B7%29%28x%2B1%29.



Note: you can check the answer by expanding %28x%2B7%29%28x%2B1%29 to get x%5E2%2B8%2Ax%2B7 or by graphing the original expression and the answer (the two graphs should be identical).