document.write( "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? \n" ); document.write( "

Algebra.Com's Answer #306259 by rwm(914) $\"\"$ $\"About$

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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 $\"-23\"$ :

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First Number	Second Number	Sum
1	42	1+42=43
2	21	2+21=23
3	14	3+14=17
6	7	6+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 $\"7x-2\"$

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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).

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