document.write( "Question 640408: Could you help me with this problem about factoring polynomials completely\r
\n" ); document.write( "\n" ); document.write( "4x^2-20x+25\r
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Algebra.Com's Answer #403222 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"4x%5E2-20x%2B25\"$ , we can see that the first coefficient is $\"4\"$ , the second coefficient is $\"-20\"$ , and the last term is $\"25\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"4\"$ by the last term $\"25\"$ to get $\"%284%29%2825%29=100\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"100\"$ (the previous product) and add to the second coefficient $\"-20\"$ ?\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 $\"100\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"100\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,4,5,10,20,25,50,100\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-4,-5,-10,-20,-25,-50,-100\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 $\"100\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*100 = 100
\n" ); document.write( "2*50 = 100
\n" ); document.write( "4*25 = 100
\n" ); document.write( "5*20 = 100
\n" ); document.write( "10*10 = 100
\n" ); document.write( "(-1)*(-100) = 100
\n" ); document.write( "(-2)*(-50) = 100
\n" ); document.write( "(-4)*(-25) = 100
\n" ); document.write( "(-5)*(-20) = 100
\n" ); document.write( "(-10)*(-10) = 100\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 $\"-20\"$ :\r
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First Number	Second Number	Sum
1	100	1+100=101
2	50	2+50=52
4	25	4+25=29
5	20	5+20=25
10	10	10+10=20
-1	-100	-1+(-100)=-101
-2	-50	-2+(-50)=-52
-4	-25	-4+(-25)=-29
-5	-20	-5+(-20)=-25
-10	-10	-10+(-10)=-20

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-10\"$ and $\"-10\"$ add to $\"-20\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-10\"$ and $\"-10\"$ both multiply to $\"100\"$ and add to $\"-20\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-20x\"$ with $\"-10x-10x\"$ . Remember, $\"-10\"$ and $\"-10\"$ add to $\"-20\"$ . So this shows us that $\"-10x-10x=-20x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"4x%5E2%2Bhighlight%28-10x-10x%29%2B25\"$ Replace the second term $\"-20x\"$ with $\"-10x-10x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%284x%5E2-10x%29%2B%28-10x%2B25%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"2x%282x-5%29%2B%28-10x%2B25%29\"$ Factor out the GCF $\"2x\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"2x%282x-5%29-5%282x-5%29\"$ Factor out $\"5\"$ 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( " $\"%282x-5%29%282x-5%29\"$ Combine like terms. Or factor out the common term $\"2x-5\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"%282x-5%29%5E2\"$ Condense the terms.\r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"4x%5E2-20x%2B25\"$ factors to $\"%282x-5%29%5E2\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"4x%5E2-20x%2B25=%282x-5%29%5E2\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%282x-5%29%5E2\"$ to get $\"4x%5E2-20x%2B25\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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