document.write( "Question 778421: Factor.\r
\n" ); document.write( "\n" ); document.write( "−3y^3 + 15y^2 − 18y \n" ); document.write( "

Algebra.Com's Answer #474632 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( " $\"-3y%5E3%2B15y%5E2-18y\"$ Start with the given expression.\r
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\n" ); document.write( "\n" ); document.write( " $\"-3y%28y%5E2-5y%2B6%29\"$ Factor out the GCF $\"-3y\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now let's try to factor the inner expression $\"y%5E2-5y%2B6\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"y%5E2-5y%2B6\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"-5\"$ , and the last term is $\"6\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"1\"$ by the last term $\"6\"$ to get $\"%281%29%286%29=6\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"6\"$ (the previous product) and add to the second coefficient $\"-5\"$ ?\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 $\"6\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"6\"$ :\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 $\"6\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*6 = 6
\n" ); document.write( "2*3 = 6
\n" ); document.write( "(-1)*(-6) = 6
\n" ); document.write( "(-2)*(-3) = 6\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 $\"-5\"$ :\r
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First Number	Second Number	Sum
1	6	1+6=7
2	3	2+3=5
-1	-6	-1+(-6)=-7
-2	-3	-2+(-3)=-5

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-2\"$ and $\"-3\"$ add to $\"-5\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-2\"$ and $\"-3\"$ both multiply to $\"6\"$ and add to $\"-5\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-5y\"$ with $\"-2y-3y\"$ . Remember, $\"-2\"$ and $\"-3\"$ add to $\"-5\"$ . So this shows us that $\"-2y-3y=-5y\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"y%5E2%2Bhighlight%28-2y-3y%29%2B6\"$ Replace the second term $\"-5y\"$ with $\"-2y-3y\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%28y%5E2-2y%29%2B%28-3y%2B6%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"y%28y-2%29%2B%28-3y%2B6%29\"$ Factor out the GCF $\"y\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"y%28y-2%29-3%28y-2%29\"$ 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.\r
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\n" ); document.write( "\n" ); document.write( " $\"%28y-3%29%28y-2%29\"$ Combine like terms. Or factor out the common term $\"y-2\"$ \r
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\n" ); document.write( "\n" ); document.write( "So $\"-3y%28y%5E2-5y%2B6%29\"$ then factors further to $\"-3y%28y-3%29%28y-2%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"-3y%5E3%2B15y%5E2-18y\"$ completely factors to $\"-3y%28y-3%29%28y-2%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"-3y%5E3%2B15y%5E2-18y=-3y%28y-3%29%28y-2%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"-3y%28y-3%29%28y-2%29\"$ to get $\"-3y%5E3%2B15y%5E2-18y\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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