SOLUTION: What are the missing dimensions of a solid rectangle if the volume is x^3+5x^2+8x+4 and the height is x+2? Hi. I'm Hannah. Need help with this please. Thank you!

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Question 889244: What are the missing dimensions of a solid rectangle if the volume is x^3+5x^2+8x+4 and the height is x+2?
Hi. I'm Hannah. Need help with this please.
Thank you!



Found 2 solutions by josgarithmetic, richwmiller:
Answer by josgarithmetic(39617) About Me  (Show Source):
You can put this solution on YOUR website!
The missing dimensions are length and width, the names depending on how you choose to use the terminology.

Divide the volume expression by the height expression. You should be able to factorize the quotient. Each of these two binomial factors is one of each of the dimensions you want.

%28x%5E3%2B5x%5E2%2B8x%2B4%29%2F%28x%2B2%29=highlight_green%28x%5E2%2B3x%2B2%29, using synthetic division, although polynomial division gives the same.

Answer by richwmiller(17219) About Me  (Show Source):
You can put this solution on YOUR website!
x^3+5x^2+8 x+4 = (x^2+3x+2) × (x+2)
(x^2+3x+2)
Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)


Looking at the expression x%5E2%2B3x%2B2, we can see that the first coefficient is 1, the second coefficient is 3, and the last term is 2.



Now multiply the first coefficient 1 by the last term 2 to get %281%29%282%29=2.



Now the question is: what two whole numbers multiply to 2 (the previous product) and add to the second coefficient 3?



To find these two numbers, we need to list all of the factors of 2 (the previous product).



Factors of 2:

1,2

-1,-2



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



These factors pair up and multiply to 2.

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


Now let's add up each pair of factors to see if one pair adds to the middle coefficient 3:



First NumberSecond NumberSum
121+2=3
-1-2-1+(-2)=-3




From the table, we can see that the two numbers 1 and 2 add to 3 (the middle coefficient).



So the two numbers 1 and 2 both multiply to 2 and add to 3



Now replace the middle term 3x with x%2B2x. Remember, 1 and 2 add to 3. So this shows us that x%2B2x=3x.



x%5E2%2Bhighlight%28x%2B2x%29%2B2 Replace the second term 3x with x%2B2x.



%28x%5E2%2Bx%29%2B%282x%2B2%29 Group the terms into two pairs.



x%28x%2B1%29%2B%282x%2B2%29 Factor out the GCF x from the first group.



x%28x%2B1%29%2B2%28x%2B1%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.



%28x%2B2%29%28x%2B1%29 Combine like terms. Or factor out the common term x%2B1



===============================================================



Answer:



So x%5E2%2B3%2Ax%2B2 factors to %28x%2B2%29%28x%2B1%29.



In other words, x%5E2%2B3%2Ax%2B2=%28x%2B2%29%28x%2B1%29.



Note: you can check the answer by expanding %28x%2B2%29%28x%2B1%29 to get x%5E2%2B3%2Ax%2B2 or by graphing the original expression and the answer (the two graphs should be identical).


so the three dimensions are
x+2
x+2
x+1