document.write( "Question 635185: I need help factoring y^3-14y^2+45y \n" ); document.write( "

Algebra.Com's Answer #400146 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( " $\"y%5E3-14y%5E2%2B45y\"$ Start with the given expression.\r
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\n" ); document.write( "\n" ); document.write( " $\"y%28y%5E2-14y%2B45%29\"$ Factor out the GCF $\"y\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now let's try to factor the inner expression $\"y%5E2-14y%2B45\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"y%5E2-14y%2B45\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"-14\"$ , and the last term is $\"45\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"1\"$ by the last term $\"45\"$ to get $\"%281%29%2845%29=45\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"45\"$ (the previous product) and add to the second coefficient $\"-14\"$ ?\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 $\"45\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"45\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,3,5,9,15,45\r
\n" ); document.write( "\n" ); document.write( "-1,-3,-5,-9,-15,-45\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 $\"45\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*45 = 45
\n" ); document.write( "3*15 = 45
\n" ); document.write( "5*9 = 45
\n" ); document.write( "(-1)*(-45) = 45
\n" ); document.write( "(-3)*(-15) = 45
\n" ); document.write( "(-5)*(-9) = 45\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 $\"-14\"$ :\r
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First Number	Second Number	Sum
1	45	1+45=46
3	15	3+15=18
5	9	5+9=14
-1	-45	-1+(-45)=-46
-3	-15	-3+(-15)=-18
-5	-9	-5+(-9)=-14

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-5\"$ and $\"-9\"$ add to $\"-14\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-5\"$ and $\"-9\"$ both multiply to $\"45\"$ and add to $\"-14\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-14y\"$ with $\"-5y-9y\"$ . Remember, $\"-5\"$ and $\"-9\"$ add to $\"-14\"$ . So this shows us that $\"-5y-9y=-14y\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"y%5E2%2Bhighlight%28-5y-9y%29%2B45\"$ Replace the second term $\"-14y\"$ with $\"-5y-9y\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%28y%5E2-5y%29%2B%28-9y%2B45%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"y%28y-5%29%2B%28-9y%2B45%29\"$ Factor out the GCF $\"y\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"y%28y-5%29-9%28y-5%29\"$ 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( " $\"%28y-9%29%28y-5%29\"$ Combine like terms. Or factor out the common term $\"y-5\"$ \r
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\n" ); document.write( "\n" ); document.write( "So $\"y%28y%5E2-14y%2B45%29\"$ then factors further to $\"y%28y-9%29%28y-5%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"y%5E3-14y%5E2%2B45y\"$ completely factors to $\"y%28y-9%29%28y-5%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"y%5E3-14y%5E2%2B45y=y%28y-9%29%28y-5%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"y%28y-9%29%28y-5%29\"$ to get $\"y%5E3-14y%5E2%2B45y\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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