document.write( "Question 173380: I just want to make sure I
\n" ); document.write( "'m on the right track.Factoring\r
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Algebra.Com's Answer #128225 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
# 1\r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"x%5E2-5x%2B24\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"-5\"$ , and the last term is $\"24\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"1\"$ by the last term $\"24\"$ to get $\"%281%29%2824%29=24\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"24\"$ (the previous product) and add to the second coefficient $\"-5\"$ ?\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 $\"24\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"24\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,6,8,12,24\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-6,-8,-12,-24\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 $\"24\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*24
\n" ); document.write( "2*12
\n" ); document.write( "3*8
\n" ); document.write( "4*6
\n" ); document.write( "(-1)*(-24)
\n" ); document.write( "(-2)*(-12)
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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 $\"-5\"$ :\r
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First Number	Second Number	Sum
1	24	1+24=25
2	12	2+12=14
3	8	3+8=11
4	6	4+6=10
-1	-24	-1+(-24)=-25
-2	-12	-2+(-12)=-14
-3	-8	-3+(-8)=-11
-4	-6	-4+(-6)=-10

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that there are no pairs of numbers which add to $\"-5\"$ . So $\"x%5E2-5x%2B24\"$ cannot be factored.\r
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\n" ); document.write( "\n" ); document.write( "# 2\r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"x%5E2%2B3x-40\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"3\"$ , and the last term is $\"-40\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"1\"$ by the last term $\"-40\"$ to get $\"%281%29%28-40%29=-40\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-40\"$ (the previous product) and add to the second coefficient $\"3\"$ ?\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 $\"-40\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"-40\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,4,5,8,10,20,40\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-4,-5,-8,-10,-20,-40\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 $\"-40\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-40)
\n" ); document.write( "2*(-20)
\n" ); document.write( "4*(-10)
\n" ); document.write( "5*(-8)
\n" ); document.write( "(-1)*(40)
\n" ); document.write( "(-2)*(20)
\n" ); document.write( "(-4)*(10)
\n" ); document.write( "(-5)*(8)\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 $\"3\"$ :\r
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First Number	Second Number	Sum
1	-40	1+(-40)=-39
2	-20	2+(-20)=-18
4	-10	4+(-10)=-6
5	-8	5+(-8)=-3
-1	40	-1+40=39
-2	20	-2+20=18
-4	10	-4+10=6
-5	8	-5+8=3

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-5\"$ and $\"8\"$ add to $\"3\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-5\"$ and $\"8\"$ both multiply to $\"-40\"$ and add to $\"3\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"3x\"$ with $\"-5x%2B8x\"$ . Remember, $\"-5\"$ and $\"8\"$ add to $\"3\"$ . So this shows us that $\"-5x%2B8x=3x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"x%5E2%2Bhighlight%28-5x%2B8x%29-40\"$ Replace the second term $\"3x\"$ with $\"-5x%2B8x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%28x%5E2-5x%29%2B%288x-40%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%28x-5%29%2B%288x-40%29\"$ Factor out the GCF $\"x\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%28x-5%29%2B8%28x-5%29\"$ Factor out $\"8\"$ 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( " $\"%28x%2B8%29%28x-5%29\"$ Combine like terms. Or factor out the common term $\"x-5\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"x%5E2%2B3x-40\"$ factors to $\"%28x%2B8%29%28x-5%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by FOILing $\"%28x%2B8%29%28x-5%29\"$ to get $\"x%5E2%2B3x-40\"$ or by graphing the original expression and the answer (the two graphs should be identical).\r
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\n" ); document.write( "\n" ); document.write( "# 3\r
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\n" ); document.write( "\n" ); document.write( " $\"x%5E2-9%2F16\"$ Start with the given expression.\r
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\n" ); document.write( "\n" ); document.write( " $\"%28x%29%5E2-9%2F16\"$ Rewrite $\"x%5E2\"$ as $\"%28x%29%5E2\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%28x%29%5E2-%283%2F4%29%5E2\"$ Rewrite $\"9%2F16\"$ as $\"%283%2F4%29%5E2\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Notice how we have a difference of squares $\"A%5E2-B%5E2\"$ where in this case $\"A=x\"$ and $\"B=3%2F4\"$ .\r
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\n" ); document.write( "\n" ); document.write( "So let's use the difference of squares formula $\"A%5E2-B%5E2=%28A%2BB%29%28A-B%29\"$ to factor the expression:\r
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\n" ); document.write( "\n" ); document.write( " $\"A%5E2-B%5E2=%28A%2BB%29%28A-B%29\"$ Start with the difference of squares formula.\r
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\n" ); document.write( "\n" ); document.write( " $\"%28x%29%5E2-%283%2F4%29%5E2=%28x%2B3%2F4%29%28x-3%2F4%29\"$ Plug in $\"A=x\"$ and $\"B=3%2F4\"$ .\r
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\n" ); document.write( "\n" ); document.write( "So this shows us that $\"x%5E2-9%2F16\"$ factors to $\"%28x%2B3%2F4%29%28x-3%2F4%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words $\"x%5E2-9%2F16=%28x%2B3%2F4%29%28x-3%2F4%29\"$ .
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