SOLUTION: Factor completely. Remember to look first for a common factor and to check by multiplying. If a polynomial is prime, state this. {{{4d^2+16d+16}}}

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: Factor completely. Remember to look first for a common factor and to check by multiplying. If a polynomial is prime, state this. {{{4d^2+16d+16}}}      Log On


   

Question 892664: Factor completely. Remember to look first for a common factor and to check by multiplying. If a polynomial is prime, state this.
4d%5E2%2B16d%2B16
Answer by richwmiller(17219) About Me  (Show Source):
You can put this solution on YOUR website!
Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)


4%2Ad%5E2%2B16%2Ad%2B16 Start with the given expression.



4%28d%5E2%2B4d%2B4%29 Factor out the GCF 4.



Now let's try to factor the inner expression d%5E2%2B4d%2B4



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Looking at the expression d%5E2%2B4d%2B4, we can see that the first coefficient is 1, the second coefficient is 4, and the last term is 4.



Now multiply the first coefficient 1 by the last term 4 to get %281%29%284%29=4.



Now the question is: what two whole numbers multiply to 4 (the previous product) and add to the second coefficient 4?



To find these two numbers, we need to list all of the factors of 4 (the previous product).



Factors of 4:

1,2,4

-1,-2,-4



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



These factors pair up and multiply to 4.

1*4 = 4
2*2 = 4
(-1)*(-4) = 4
(-2)*(-2) = 4


Now let's add up each pair of factors to see if one pair adds to the middle coefficient 4:



First NumberSecond NumberSum
141+4=5
222+2=4
-1-4-1+(-4)=-5
-2-2-2+(-2)=-4




From the table, we can see that the two numbers 2 and 2 add to 4 (the middle coefficient).



So the two numbers 2 and 2 both multiply to 4 and add to 4



Now replace the middle term 4d with 2d%2B2d. Remember, 2 and 2 add to 4. So this shows us that 2d%2B2d=4d.



d%5E2%2Bhighlight%282d%2B2d%29%2B4 Replace the second term 4d with 2d%2B2d.



%28d%5E2%2B2d%29%2B%282d%2B4%29 Group the terms into two pairs.



d%28d%2B2%29%2B%282d%2B4%29 Factor out the GCF d from the first group.



d%28d%2B2%29%2B2%28d%2B2%29 Factor out 2 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.



%28d%2B2%29%28d%2B2%29 Combine like terms. Or factor out the common term d%2B2



%28d%2B2%29%5E2 Condense the terms.



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So 4%28d%5E2%2B4d%2B4%29 then factors further to 4%28d%2B2%29%5E2



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Answer:



So 4%2Ad%5E2%2B16%2Ad%2B16 completely factors to 4%28d%2B2%29%5E2.



In other words, 4%2Ad%5E2%2B16%2Ad%2B16=4%28d%2B2%29%5E2.



Note: you can check the answer by expanding 4%28d%2B2%29%5E2 to get 4%2Ad%5E2%2B16%2Ad%2B16 or by graphing the original expression and the answer (the two graphs should be identical).