SOLUTION: Tim has a rectangular garden whose area can be expressed by the polynomial 7x^2-23x+6. What are the possible dimensions for the length and width?

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: Tim has a rectangular garden whose area can be expressed by the polynomial 7x^2-23x+6. What are the possible dimensions for the length and width?      Log On


   

Question 444154: Tim has a rectangular garden whose area can be expressed by the polynomial 7x^2-23x+6. What are the possible dimensions for the length and width?
Answer by rwm(914) 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 7x%5E2-23x%2B6, we can see that the first coefficient is 7, the second coefficient is -23, and the last term is 6.



Now multiply the first coefficient 7 by the last term 6 to get %287%29%286%29=42.



Now the question is: what two whole numbers multiply to 42 (the previous product) and add to the second coefficient -23?



To find these two numbers, we need to list all of the factors of 42 (the previous product).



Factors of 42:

1,2,3,6,7,14,21,42

-1,-2,-3,-6,-7,-14,-21,-42



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



These factors pair up and multiply to 42.

1*42 = 42
2*21 = 42
3*14 = 42
6*7 = 42
(-1)*(-42) = 42
(-2)*(-21) = 42
(-3)*(-14) = 42
(-6)*(-7) = 42


Now let's add up each pair of factors to see if one pair adds to the middle coefficient -23:



First NumberSecond NumberSum
1421+42=43
2212+21=23
3143+14=17
676+7=13
-1-42-1+(-42)=-43
-2-21-2+(-21)=-23
-3-14-3+(-14)=-17
-6-7-6+(-7)=-13




From the table, we can see that the two numbers -2 and -21 add to -23 (the middle coefficient).



So the two numbers -2 and -21 both multiply to 42 and add to -23



Now replace the middle term -23x with -2x-21x. Remember, -2 and -21 add to -23. So this shows us that -2x-21x=-23x.



7x%5E2%2Bhighlight%28-2x-21x%29%2B6 Replace the second term -23x with -2x-21x.



%287x%5E2-2x%29%2B%28-21x%2B6%29 Group the terms into two pairs.



x%287x-2%29%2B%28-21x%2B6%29 Factor out the GCF x from the first group.



x%287x-2%29-3%287x-2%29 Factor out 3 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-3%29%287x-2%29 Combine like terms. Or factor out the common term 7x-2



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



Answer:



So 7%2Ax%5E2-23%2Ax%2B6 factors to %28x-3%29%287x-2%29.



In other words, 7%2Ax%5E2-23%2Ax%2B6=%28x-3%29%287x-2%29.



Note: you can check the answer by expanding %28x-3%29%287x-2%29 to get 7%2Ax%5E2-23%2Ax%2B6 or by graphing the original expression and the answer (the two graphs should be identical).