document.write( "Question 599740: What is the factor to
\n" ); document.write( " b squared + 16b + 63 ? \n" ); document.write( "

Algebra.Com's Answer #379059 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"b%5E2%2B16b%2B63\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"16\"$ , and the last term is $\"63\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"1\"$ by the last term $\"63\"$ to get $\"%281%29%2863%29=63\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"63\"$ (the previous product) and add to the second coefficient $\"16\"$ ?\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 $\"63\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"63\"$ :\r
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\n" ); document.write( "\n" ); document.write( "-1,-3,-7,-9,-21,-63\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 $\"63\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*63 = 63
\n" ); document.write( "3*21 = 63
\n" ); document.write( "7*9 = 63
\n" ); document.write( "(-1)*(-63) = 63
\n" ); document.write( "(-3)*(-21) = 63
\n" ); document.write( "(-7)*(-9) = 63\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 $\"16\"$ :\r
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First Number	Second Number	Sum
1	63	1+63=64
3	21	3+21=24
7	9	7+9=16
-1	-63	-1+(-63)=-64
-3	-21	-3+(-21)=-24
-7	-9	-7+(-9)=-16

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"7\"$ and $\"9\"$ add to $\"16\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"7\"$ and $\"9\"$ both multiply to $\"63\"$ and add to $\"16\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"16b\"$ with $\"7b%2B9b\"$ . Remember, $\"7\"$ and $\"9\"$ add to $\"16\"$ . So this shows us that $\"7b%2B9b=16b\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"b%5E2%2Bhighlight%287b%2B9b%29%2B63\"$ Replace the second term $\"16b\"$ with $\"7b%2B9b\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%28b%5E2%2B7b%29%2B%289b%2B63%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"b%28b%2B7%29%2B%289b%2B63%29\"$ Factor out the GCF $\"b\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"b%28b%2B7%29%2B9%28b%2B7%29\"$ Factor out $\"9\"$ 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( " $\"%28b%2B9%29%28b%2B7%29\"$ Combine like terms. Or factor out the common term $\"b%2B7\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"b%5E2%2B16b%2B63\"$ factors to $\"%28b%2B9%29%28b%2B7%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"b%5E2%2B16b%2B63=%28b%2B9%29%28b%2B7%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%28b%2B9%29%28b%2B7%29\"$ to get $\"b%5E2%2B16b%2B63\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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