document.write( "Question 610503: how do you factor r squared+15r+56 completely? \n" ); document.write( "

Algebra.Com's Answer #384426 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"r%5E2%2B15r%2B56\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"15\"$ , and the last term is $\"56\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"1\"$ by the last term $\"56\"$ to get $\"%281%29%2856%29=56\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"56\"$ (the previous product) and add to the second coefficient $\"15\"$ ?\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 $\"56\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"56\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,4,7,8,14,28,56\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-4,-7,-8,-14,-28,-56\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 $\"56\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*56 = 56
\n" ); document.write( "2*28 = 56
\n" ); document.write( "4*14 = 56
\n" ); document.write( "7*8 = 56
\n" ); document.write( "(-1)*(-56) = 56
\n" ); document.write( "(-2)*(-28) = 56
\n" ); document.write( "(-4)*(-14) = 56
\n" ); document.write( "(-7)*(-8) = 56\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 $\"15\"$ :\r
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First Number	Second Number	Sum
1	56	1+56=57
2	28	2+28=30
4	14	4+14=18
7	8	7+8=15
-1	-56	-1+(-56)=-57
-2	-28	-2+(-28)=-30
-4	-14	-4+(-14)=-18
-7	-8	-7+(-8)=-15

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"7\"$ and $\"8\"$ add to $\"15\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"7\"$ and $\"8\"$ both multiply to $\"56\"$ and add to $\"15\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"15r\"$ with $\"7r%2B8r\"$ . Remember, $\"7\"$ and $\"8\"$ add to $\"15\"$ . So this shows us that $\"7r%2B8r=15r\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"r%5E2%2Bhighlight%287r%2B8r%29%2B56\"$ Replace the second term $\"15r\"$ with $\"7r%2B8r\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%28r%5E2%2B7r%29%2B%288r%2B56%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"r%28r%2B7%29%2B%288r%2B56%29\"$ Factor out the GCF $\"r\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"r%28r%2B7%29%2B8%28r%2B7%29\"$ Factor out $\"8\"$ 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( " $\"%28r%2B8%29%28r%2B7%29\"$ Combine like terms. Or factor out the common term $\"r%2B7\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"r%5E2%2B15r%2B56\"$ factors to $\"%28r%2B8%29%28r%2B7%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"r%5E2%2B15r%2B56=%28r%2B8%29%28r%2B7%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%28r%2B8%29%28r%2B7%29\"$ to get $\"r%5E2%2B15r%2B56\"$ or by graphing the original expression and the answer (the two graphs should be identical).\r
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