document.write( "Question 615983: factor the trinomial 24r^2-6r-45 \n" ); document.write( "

Algebra.Com's Answer #387425 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( " $\"24r%5E2-6r-45\"$ Start with the given expression.\r
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\n" ); document.write( "\n" ); document.write( " $\"3%288r%5E2-2r-15%29\"$ Factor out the GCF $\"3\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now let's try to factor the inner expression $\"8r%5E2-2r-15\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"8r%5E2-2r-15\"$ , we can see that the first coefficient is $\"8\"$ , the second coefficient is $\"-2\"$ , and the last term is $\"-15\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"8\"$ by the last term $\"-15\"$ to get $\"%288%29%28-15%29=-120\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-120\"$ (the previous product) and add to the second coefficient $\"-2\"$ ?\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 $\"-120\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"-120\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,5,6,8,10,12,15,20,24,30,40,60,120\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-5,-6,-8,-10,-12,-15,-20,-24,-30,-40,-60,-120\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 $\"-120\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-120) = -120
\n" ); document.write( "2*(-60) = -120
\n" ); document.write( "3*(-40) = -120
\n" ); document.write( "4*(-30) = -120
\n" ); document.write( "5*(-24) = -120
\n" ); document.write( "6*(-20) = -120
\n" ); document.write( "8*(-15) = -120
\n" ); document.write( "10*(-12) = -120
\n" ); document.write( "(-1)*(120) = -120
\n" ); document.write( "(-2)*(60) = -120
\n" ); document.write( "(-3)*(40) = -120
\n" ); document.write( "(-4)*(30) = -120
\n" ); document.write( "(-5)*(24) = -120
\n" ); document.write( "(-6)*(20) = -120
\n" ); document.write( "(-8)*(15) = -120
\n" ); document.write( "(-10)*(12) = -120\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 $\"-2\"$ :\r
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First Number	Second Number	Sum
1	-120	1+(-120)=-119
2	-60	2+(-60)=-58
3	-40	3+(-40)=-37
4	-30	4+(-30)=-26
5	-24	5+(-24)=-19
6	-20	6+(-20)=-14
8	-15	8+(-15)=-7
10	-12	10+(-12)=-2
-1	120	-1+120=119
-2	60	-2+60=58
-3	40	-3+40=37
-4	30	-4+30=26
-5	24	-5+24=19
-6	20	-6+20=14
-8	15	-8+15=7
-10	12	-10+12=2

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"10\"$ and $\"-12\"$ add to $\"-2\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"10\"$ and $\"-12\"$ both multiply to $\"-120\"$ and add to $\"-2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-2r\"$ with $\"10r-12r\"$ . Remember, $\"10\"$ and $\"-12\"$ add to $\"-2\"$ . So this shows us that $\"10r-12r=-2r\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"8r%5E2%2Bhighlight%2810r-12r%29-15\"$ Replace the second term $\"-2r\"$ with $\"10r-12r\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%288r%5E2%2B10r%29%2B%28-12r-15%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"2r%284r%2B5%29%2B%28-12r-15%29\"$ Factor out the GCF $\"2r\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"2r%284r%2B5%29-3%284r%2B5%29\"$ 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( " $\"%282r-3%29%284r%2B5%29\"$ Combine like terms. Or factor out the common term $\"4r%2B5\"$ \r
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\n" ); document.write( "\n" ); document.write( "So $\"3%288r%5E2-2r-15%29\"$ then factors further to $\"3%282r-3%29%284r%2B5%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"24r%5E2-6r-45\"$ completely factors to $\"3%282r-3%29%284r%2B5%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"24r%5E2-6r-45=3%282r-3%29%284r%2B5%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"3%282r-3%29%284r%2B5%29\"$ to get $\"24r%5E2-6r-45\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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