document.write( "Question 925883: 8c^2 + 38c + 35 factor \n" ); document.write( "

Algebra.Com's Answer #561897 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"8c%5E2%2B38c%2B35\"$ , we can see that the first coefficient is $\"8\"$ , the second coefficient is $\"38\"$ , and the last term is $\"35\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"8\"$ by the last term $\"35\"$ to get $\"%288%29%2835%29=280\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"280\"$ (the previous product) and add to the second coefficient $\"38\"$ ?\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 $\"280\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"280\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,4,5,7,8,10,14,20,28,35,40,56,70,140,280\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-4,-5,-7,-8,-10,-14,-20,-28,-35,-40,-56,-70,-140,-280\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 $\"280\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*280 = 280
\n" ); document.write( "2*140 = 280
\n" ); document.write( "4*70 = 280
\n" ); document.write( "5*56 = 280
\n" ); document.write( "7*40 = 280
\n" ); document.write( "8*35 = 280
\n" ); document.write( "10*28 = 280
\n" ); document.write( "14*20 = 280
\n" ); document.write( "(-1)*(-280) = 280
\n" ); document.write( "(-2)*(-140) = 280
\n" ); document.write( "(-4)*(-70) = 280
\n" ); document.write( "(-5)*(-56) = 280
\n" ); document.write( "(-7)*(-40) = 280
\n" ); document.write( "(-8)*(-35) = 280
\n" ); document.write( "(-10)*(-28) = 280
\n" ); document.write( "(-14)*(-20) = 280\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 $\"38\"$ :\r
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First Number	Second Number	Sum
1	280	1+280=281
2	140	2+140=142
4	70	4+70=74
5	56	5+56=61
7	40	7+40=47
8	35	8+35=43
10	28	10+28=38
14	20	14+20=34
-1	-280	-1+(-280)=-281
-2	-140	-2+(-140)=-142
-4	-70	-4+(-70)=-74
-5	-56	-5+(-56)=-61
-7	-40	-7+(-40)=-47
-8	-35	-8+(-35)=-43
-10	-28	-10+(-28)=-38
-14	-20	-14+(-20)=-34

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"10\"$ and $\"28\"$ add to $\"38\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"10\"$ and $\"28\"$ both multiply to $\"280\"$ and add to $\"38\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"38c\"$ with $\"10c%2B28c\"$ . Remember, $\"10\"$ and $\"28\"$ add to $\"38\"$ . So this shows us that $\"10c%2B28c=38c\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"8c%5E2%2Bhighlight%2810c%2B28c%29%2B35\"$ Replace the second term $\"38c\"$ with $\"10c%2B28c\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%288c%5E2%2B10c%29%2B%2828c%2B35%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"2c%284c%2B5%29%2B%2828c%2B35%29\"$ Factor out the GCF $\"2c\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"2c%284c%2B5%29%2B7%284c%2B5%29\"$ Factor out $\"7\"$ 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( " $\"%282c%2B7%29%284c%2B5%29\"$ Combine like terms. Or factor out the common term $\"4c%2B5\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"8c%5E2%2B38c%2B35\"$ factors to $\"%282c%2B7%29%284c%2B5%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"8c%5E2%2B38c%2B35=%282c%2B7%29%284c%2B5%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%282c%2B7%29%284c%2B5%29\"$ to get $\"8c%5E2%2B38c%2B35\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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