document.write( "Question 551510: Megan factored the expression 28x^2 + x - 15 as (4x-3)(7x+5). But when Jacob applied the FOIL principle and multiplied out (4x-3)(7x+5), he got 28x^2 - x - 15; thus, Megan’s solution does not appear to check. Why is that? Please help Megan to understand this better. Explain your reasoning and correctly factor the original expression, if possible. If the expression is prime, so state. \n" ); document.write( "

Algebra.Com's Answer #359908 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"28x%5E2%2Bx-15\"$ , we can see that the first coefficient is $\"28\"$ , the second coefficient is $\"1\"$ , and the last term is $\"-15\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"28\"$ by the last term $\"-15\"$ to get $\"%2828%29%28-15%29=-420\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-420\"$ (the previous product) and add to the second coefficient $\"1\"$ ?\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 $\"-420\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"-420\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,5,6,7,10,12,14,15,20,21,28,30,35,42,60,70,84,105,140,210,420\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-5,-6,-7,-10,-12,-14,-15,-20,-21,-28,-30,-35,-42,-60,-70,-84,-105,-140,-210,-420\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 $\"-420\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-420) = -420
\n" ); document.write( "2*(-210) = -420
\n" ); document.write( "3*(-140) = -420
\n" ); document.write( "4*(-105) = -420
\n" ); document.write( "5*(-84) = -420
\n" ); document.write( "6*(-70) = -420
\n" ); document.write( "7*(-60) = -420
\n" ); document.write( "10*(-42) = -420
\n" ); document.write( "12*(-35) = -420
\n" ); document.write( "14*(-30) = -420
\n" ); document.write( "15*(-28) = -420
\n" ); document.write( "20*(-21) = -420
\n" ); document.write( "(-1)*(420) = -420
\n" ); document.write( "(-2)*(210) = -420
\n" ); document.write( "(-3)*(140) = -420
\n" ); document.write( "(-4)*(105) = -420
\n" ); document.write( "(-5)*(84) = -420
\n" ); document.write( "(-6)*(70) = -420
\n" ); document.write( "(-7)*(60) = -420
\n" ); document.write( "(-10)*(42) = -420
\n" ); document.write( "(-12)*(35) = -420
\n" ); document.write( "(-14)*(30) = -420
\n" ); document.write( "(-15)*(28) = -420
\n" ); document.write( "(-20)*(21) = -420\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 $\"1\"$ :\r
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First Number	Second Number	Sum
1	-420	1+(-420)=-419
2	-210	2+(-210)=-208
3	-140	3+(-140)=-137
4	-105	4+(-105)=-101
5	-84	5+(-84)=-79
6	-70	6+(-70)=-64
7	-60	7+(-60)=-53
10	-42	10+(-42)=-32
12	-35	12+(-35)=-23
14	-30	14+(-30)=-16
15	-28	15+(-28)=-13
20	-21	20+(-21)=-1
-1	420	-1+420=419
-2	210	-2+210=208
-3	140	-3+140=137
-4	105	-4+105=101
-5	84	-5+84=79
-6	70	-6+70=64
-7	60	-7+60=53
-10	42	-10+42=32
-12	35	-12+35=23
-14	30	-14+30=16
-15	28	-15+28=13
-20	21	-20+21=1

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-20\"$ and $\"21\"$ add to $\"1\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-20\"$ and $\"21\"$ both multiply to $\"-420\"$ and add to $\"1\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"1x\"$ with $\"-20x%2B21x\"$ . Remember, $\"-20\"$ and $\"21\"$ add to $\"1\"$ . So this shows us that $\"-20x%2B21x=1x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"28x%5E2%2Bhighlight%28-20x%2B21x%29-15\"$ Replace the second term $\"1x\"$ with $\"-20x%2B21x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%2828x%5E2-20x%29%2B%2821x-15%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"4x%287x-5%29%2B%2821x-15%29\"$ Factor out the GCF $\"4x\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"4x%287x-5%29%2B3%287x-5%29\"$ 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( " $\"%284x%2B3%29%287x-5%29\"$ Combine like terms. Or factor out the common term $\"7x-5\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"28x%5E2%2Bx-15\"$ factors to $\"%284x%2B3%29%287x-5%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"28x%5E2%2Bx-15=%284x%2B3%29%287x-5%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%284x%2B3%29%287x-5%29\"$ to get $\"28x%5E2%2Bx-15\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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