document.write( "Question 156340: 6. How do I complete this using factoring? I appreciate the help.\r
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\n" ); document.write( "\n" ); document.write( "8x^2 - 22x - 21 \n" ); document.write( "

Algebra.Com's Answer #115128 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"8x%5E2-22x-21\"$ , we can see that the first coefficient is $\"8\"$ , the second coefficient is $\"-22\"$ , and the last term is $\"-21\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"8\"$ by the last term $\"-21\"$ to get $\"%288%29%28-21%29=-168\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-168\"$ (the previous product) and add to the second coefficient $\"-22\"$ ?\r
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\n" ); document.write( "\n" ); document.write( "To find these two numbers, we need to list all of the factors of $\"-168\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"-168\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,6,7,8,12,14,21,24,28,42,56,84,168\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-6,-7,-8,-12,-14,-21,-24,-28,-42,-56,-84,-168\r
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\n" ); document.write( "\n" ); document.write( "Note: list the negative of each factor. This will allow us to find all possible combinations.\r
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\n" ); document.write( "\n" ); document.write( "These factors pair up and multiply to $\"-168\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-168)
\n" ); document.write( "2*(-84)
\n" ); document.write( "3*(-56)
\n" ); document.write( "4*(-42)
\n" ); document.write( "6*(-28)
\n" ); document.write( "7*(-24)
\n" ); document.write( "8*(-21)
\n" ); document.write( "12*(-14)
\n" ); document.write( "(-1)*(168)
\n" ); document.write( "(-2)*(84)
\n" ); document.write( "(-3)*(56)
\n" ); document.write( "(-4)*(42)
\n" ); document.write( "(-6)*(28)
\n" ); document.write( "(-7)*(24)
\n" ); document.write( "(-8)*(21)
\n" ); document.write( "(-12)*(14)\r
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\n" ); document.write( "\n" ); document.write( "Now let's add up each pair of factors to see if one pair adds to the middle coefficient $\"-22\"$ :\r
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First Number	Second Number	Sum
1	-168	1+(-168)=-167
2	-84	2+(-84)=-82
3	-56	3+(-56)=-53
4	-42	4+(-42)=-38
6	-28	6+(-28)=-22
7	-24	7+(-24)=-17
8	-21	8+(-21)=-13
12	-14	12+(-14)=-2
-1	168	-1+168=167
-2	84	-2+84=82
-3	56	-3+56=53
-4	42	-4+42=38
-6	28	-6+28=22
-7	24	-7+24=17
-8	21	-8+21=13
-12	14	-12+14=2

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"6\"$ and $\"-28\"$ add to $\"-22\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"6\"$ and $\"-28\"$ both multiply to $\"-168\"$ and add to $\"-22\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-22x\"$ with $\"6x-28x\"$ . Remember, $\"6\"$ and $\"-28\"$ add to $\"-22\"$ . So this shows us that $\"6x-28x=-22x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"8x%5E2%2Bhighlight%286x-28x%29-21\"$ Replace the second term $\"-22x\"$ with $\"6x-28x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%288x%5E2%2B6x%29%2B%28-28x-21%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"2x%284x%2B3%29%2B%28-28x-21%29\"$ Factor out the GCF $\"2x\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"2x%284x%2B3%29-7%284x%2B3%29\"$ Factor out $\"7\"$ 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.\r
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\n" ); document.write( "\n" ); document.write( " $\"%282x-7%29%284x%2B3%29\"$ Combine like terms. Or factor out the common term $\"4x%2B3\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"8x%5E2-22x-21\"$ factors to $\"%282x-7%29%284x%2B3%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by FOILing $\"%282x-7%29%284x%2B3%29\"$ to get $\"8x%5E2-22x-21\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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