SOLUTION: The lengths of the side of a rectangle can be represented by the factors of {{{6n^2+n-2}}}. Which expression could represent the perimeter of the rectangle? A. 2n-1 B. 3n+2 C. 5

Algebra ->  Test -> SOLUTION: The lengths of the side of a rectangle can be represented by the factors of {{{6n^2+n-2}}}. Which expression could represent the perimeter of the rectangle? A. 2n-1 B. 3n+2 C. 5      Log On


   

Question 739776: The lengths of the side of a rectangle can be represented by the factors of 6n%5E2%2Bn-2. Which expression could represent the perimeter of the rectangle?
A. 2n-1
B. 3n+2
C. 5n+1
D. 10n+2
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)


Looking at the expression 6n%5E2%2Bn-2, we can see that the first coefficient is 6, the second coefficient is 1, and the last term is -2.



Now multiply the first coefficient 6 by the last term -2 to get %286%29%28-2%29=-12.



Now the question is: what two whole numbers multiply to -12 (the previous product) and add to the second coefficient 1?



To find these two numbers, we need to list all of the factors of -12 (the previous product).



Factors of -12:

1,2,3,4,6,12

-1,-2,-3,-4,-6,-12



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



These factors pair up and multiply to -12.

1*(-12) = -12
2*(-6) = -12
3*(-4) = -12
(-1)*(12) = -12
(-2)*(6) = -12
(-3)*(4) = -12


Now let's add up each pair of factors to see if one pair adds to the middle coefficient 1:



First NumberSecond NumberSum
1-121+(-12)=-11
2-62+(-6)=-4
3-43+(-4)=-1
-112-1+12=11
-26-2+6=4
-34-3+4=1




From the table, we can see that the two numbers -3 and 4 add to 1 (the middle coefficient).



So the two numbers -3 and 4 both multiply to -12 and add to 1



Now replace the middle term 1n with -3n%2B4n. Remember, -3 and 4 add to 1. So this shows us that -3n%2B4n=1n.



6n%5E2%2Bhighlight%28-3n%2B4n%29-2 Replace the second term 1n with -3n%2B4n.



%286n%5E2-3n%29%2B%284n-2%29 Group the terms into two pairs.



3n%282n-1%29%2B%284n-2%29 Factor out the GCF 3n from the first group.



3n%282n-1%29%2B2%282n-1%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.



%283n%2B2%29%282n-1%29 Combine like terms. Or factor out the common term 2n-1



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Answer:



So 6%2An%5E2%2Bn-2 factors to %283n%2B2%29%282n-1%29.



In other words, 6%2An%5E2%2Bn-2=%283n%2B2%29%282n-1%29.



Note: you can check the answer by expanding %283n%2B2%29%282n-1%29 to get 6%2An%5E2%2Bn-2 or by graphing the original expression and the answer (the two graphs should be identical).


the perimeter will be 2 times the sum of the factors
d) 10n+2