SOLUTION: Completely factor the expression. x2+10x+16

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Question 887725: Completely factor the expression.
x2+10x+16
Found 2 solutions by Alan3354, jim_thompson5910:
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
Find a pair of factors of 16 that have a sum of 10.

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

Looking at the expression x%5E2%2B10x%2B16, we can see that the first coefficient is 1, the second coefficient is 10, and the last term is 16.


Now multiply the first coefficient 1 by the last term 16 to get %281%29%2816%29=16.


Now the question is: what two whole numbers multiply to 16 (the previous product) and add to the second coefficient 10?


To find these two numbers, we need to list all of the factors of 16 (the previous product).


Factors of 16:
1,2,4,8,16
-1,-2,-4,-8,-16


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


These factors pair up and multiply to 16.
1*16 = 16
2*8 = 16
4*4 = 16
(-1)*(-16) = 16
(-2)*(-8) = 16
(-4)*(-4) = 16

Now let's add up each pair of factors to see if one pair adds to the middle coefficient 10:


First NumberSecond NumberSum
1161+16=17
282+8=10
444+4=8
-1-16-1+(-16)=-17
-2-8-2+(-8)=-10
-4-4-4+(-4)=-8



From the table, we can see that the two numbers 2 and 8 add to 10 (the middle coefficient).


So the two numbers 2 and 8 both multiply to 16 and add to 10


Now replace the middle term 10x with 2x%2B8x. Remember, 2 and 8 add to 10. So this shows us that 2x%2B8x=10x.


x%5E2%2Bhighlight%282x%2B8x%29%2B16 Replace the second term 10x with 2x%2B8x.


%28x%5E2%2B2x%29%2B%288x%2B16%29 Group the terms into two pairs.


x%28x%2B2%29%2B%288x%2B16%29 Factor out the GCF x from the first group.


x%28x%2B2%29%2B8%28x%2B2%29 Factor out 8 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%2B8%29%28x%2B2%29 Combine like terms. Or factor out the common term x%2B2


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Answer:


So x%5E2%2B10x%2B16 factors to %28x%2B8%29%28x%2B2%29.


In other words, x%5E2%2B10x%2B16=%28x%2B8%29%28x%2B2%29.


Note: you can check the answer by expanding %28x%2B8%29%28x%2B2%29 to get x%5E2%2B10x%2B16 or by graphing the original expression and the answer (the two graphs should be identical).