document.write( "Question 250391: $\"220y%2B60y%5E2-225\"$
\n" ); document.write( "Factor completely. show steps. if polynomial is prime, state this.\r
\n" ); document.write( "\n" ); document.write( "I tried to do this but i don't understand how its done. My first attempt I thought it was prime but my teacher said it was wrong and it could be factored. Could you please help? \n" ); document.write( "

Algebra.Com's Answer #182307 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
$\"220y%2B60y%5E2-225\"$ Start with the given expression.\r
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\n" ); document.write( "\n" ); document.write( " $\"60y%5E2%2B220y-225\"$ Rearrange the terms.\r
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\n" ); document.write( "\n" ); document.write( " $\"60y%5E2%2B220y-225\"$ Start with the given expression.\r
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\n" ); document.write( "\n" ); document.write( " $\"5%2812y%5E2%2B44y-45%29\"$ Factor out the GCF $\"5\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now let's try to factor the inner expression $\"12y%5E2%2B44y-45\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"12y%5E2%2B44y-45\"$ , we can see that the first coefficient is $\"12\"$ , the second coefficient is $\"44\"$ , and the last term is $\"-45\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"12\"$ by the last term $\"-45\"$ to get $\"%2812%29%28-45%29=-540\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-540\"$ (the previous product) and add to the second coefficient $\"44\"$ ?\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 $\"-540\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"-540\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,5,6,9,10,12,15,18,20,27,30,36,45,54,60,90,108,135,180,270,540\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-5,-6,-9,-10,-12,-15,-18,-20,-27,-30,-36,-45,-54,-60,-90,-108,-135,-180,-270,-540\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 $\"-540\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-540) = -540
\n" ); document.write( "2*(-270) = -540
\n" ); document.write( "3*(-180) = -540
\n" ); document.write( "4*(-135) = -540
\n" ); document.write( "5*(-108) = -540
\n" ); document.write( "6*(-90) = -540
\n" ); document.write( "9*(-60) = -540
\n" ); document.write( "10*(-54) = -540
\n" ); document.write( "12*(-45) = -540
\n" ); document.write( "15*(-36) = -540
\n" ); document.write( "18*(-30) = -540
\n" ); document.write( "20*(-27) = -540
\n" ); document.write( "(-1)*(540) = -540
\n" ); document.write( "(-2)*(270) = -540
\n" ); document.write( "(-3)*(180) = -540
\n" ); document.write( "(-4)*(135) = -540
\n" ); document.write( "(-5)*(108) = -540
\n" ); document.write( "(-6)*(90) = -540
\n" ); document.write( "(-9)*(60) = -540
\n" ); document.write( "(-10)*(54) = -540
\n" ); document.write( "(-12)*(45) = -540
\n" ); document.write( "(-15)*(36) = -540
\n" ); document.write( "(-18)*(30) = -540
\n" ); document.write( "(-20)*(27) = -540\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 $\"44\"$ :\r
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First Number	Second Number	Sum
1	-540	1+(-540)=-539
2	-270	2+(-270)=-268
3	-180	3+(-180)=-177
4	-135	4+(-135)=-131
5	-108	5+(-108)=-103
6	-90	6+(-90)=-84
9	-60	9+(-60)=-51
10	-54	10+(-54)=-44
12	-45	12+(-45)=-33
15	-36	15+(-36)=-21
18	-30	18+(-30)=-12
20	-27	20+(-27)=-7
-1	540	-1+540=539
-2	270	-2+270=268
-3	180	-3+180=177
-4	135	-4+135=131
-5	108	-5+108=103
-6	90	-6+90=84
-9	60	-9+60=51
-10	54	-10+54=44
-12	45	-12+45=33
-15	36	-15+36=21
-18	30	-18+30=12
-20	27	-20+27=7

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-10\"$ and $\"54\"$ add to $\"44\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-10\"$ and $\"54\"$ both multiply to $\"-540\"$ and add to $\"44\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"44y\"$ with $\"-10y%2B54y\"$ . Remember, $\"-10\"$ and $\"54\"$ add to $\"44\"$ . So this shows us that $\"-10y%2B54y=44y\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"12y%5E2%2Bhighlight%28-10y%2B54y%29-45\"$ Replace the second term $\"44y\"$ with $\"-10y%2B54y\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%2812y%5E2-10y%29%2B%2854y-45%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"2y%286y-5%29%2B%2854y-45%29\"$ Factor out the GCF $\"2y\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"2y%286y-5%29%2B9%286y-5%29\"$ 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( " $\"%282y%2B9%29%286y-5%29\"$ Combine like terms. Or factor out the common term $\"6y-5\"$ \r
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\n" ); document.write( "\n" ); document.write( "So $\"5%2812y%5E2%2B44y-45%29\"$ then factors further to $\"5%282y%2B9%29%286y-5%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"220y%2B60y%5E2-225\"$ completely factors to $\"5%282y%2B9%29%286y-5%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"220y%2B60y%5E2-225=5%282y%2B9%29%286y-5%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"5%282y%2B9%29%286y-5%29\"$ to get $\"220y%2B60y%5E2-225\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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