document.write( "Question 670772: t^3 + 10t^2 - 24t, completely factor \n" ); document.write( "

Algebra.Com's Answer #417144 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( " $\"t%5E3%2B10t%5E2-24t\"$ Start with the given expression.\r
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\n" ); document.write( "\n" ); document.write( " $\"t%28t%5E2%2B10t-24%29\"$ Factor out the GCF $\"t\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now let's try to factor the inner expression $\"t%5E2%2B10t-24\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"t%5E2%2B10t-24\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"10\"$ , 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 $\"%281%29%28-24%29=-24\"$ .\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 $\"10\"$ ?\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) = -24
\n" ); document.write( "2*(-12) = -24
\n" ); document.write( "3*(-8) = -24
\n" ); document.write( "4*(-6) = -24
\n" ); document.write( "(-1)*(24) = -24
\n" ); document.write( "(-2)*(12) = -24
\n" ); document.write( "(-3)*(8) = -24
\n" ); document.write( "(-4)*(6) = -24\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 $\"10\"$ :\r
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First Number	Second Number	Sum
1	-24	1+(-24)=-23
2	-12	2+(-12)=-10
3	-8	3+(-8)=-5
4	-6	4+(-6)=-2
-1	24	-1+24=23
-2	12	-2+12=10
-3	8	-3+8=5
-4	6	-4+6=2

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-2\"$ and $\"12\"$ add to $\"10\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-2\"$ and $\"12\"$ both multiply to $\"-24\"$ and add to $\"10\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"10t\"$ with $\"-2t%2B12t\"$ . Remember, $\"-2\"$ and $\"12\"$ add to $\"10\"$ . So this shows us that $\"-2t%2B12t=10t\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"t%5E2%2Bhighlight%28-2t%2B12t%29-24\"$ Replace the second term $\"10t\"$ with $\"-2t%2B12t\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%28t%5E2-2t%29%2B%2812t-24%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"t%28t-2%29%2B%2812t-24%29\"$ Factor out the GCF $\"t\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"t%28t-2%29%2B12%28t-2%29\"$ Factor out $\"12\"$ 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( " $\"%28t%2B12%29%28t-2%29\"$ Combine like terms. Or factor out the common term $\"t-2\"$ \r
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\n" ); document.write( "\n" ); document.write( "So $\"t%28t%5E2%2B10t-24%29\"$ then factors further to $\"t%28t%2B12%29%28t-2%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"t%5E3%2B10t%5E2-24t\"$ completely factors to $\"t%28t%2B12%29%28t-2%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"t%5E3%2B10t%5E2-24t=t%28t%2B12%29%28t-2%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"t%28t%2B12%29%28t-2%29\"$ to get $\"t%5E3%2B10t%5E2-24t\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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