document.write( "Question 187131: please factor out polynomial.
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Algebra.Com's Answer #140271 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "\n" ); document.write( " $\"8x%5E3-14x%5E2-15x\"$ Start with the given expression\r
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\n" ); document.write( "\n" ); document.write( " $\"x%288x%5E2-14x-15%29\"$ Factor out the GCF $\"x\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's focus on the inner expression $\"8x%5E2-14x-15\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at $\"8x%5E2-14x-15\"$ we can see that the first term is $\"8x%5E2\"$ and the last term is $\"-15\"$ where the coefficients are 8 and -15 respectively.\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient 8 and the last coefficient -15 to get -120. Now what two numbers multiply to -120 and add to the middle coefficient -14? Let's list all of the factors of -120:\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
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\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-5,-6,-8,-10,-12,-15,-20,-24,-30,-40,-60,-120 ...List the negative factors as well. 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)\r
\n" ); document.write( "\n" ); document.write( "(2)*(-60)\r
\n" ); document.write( "\n" ); document.write( "(3)*(-40)\r
\n" ); document.write( "\n" ); document.write( "(4)*(-30)\r
\n" ); document.write( "\n" ); document.write( "(5)*(-24)\r
\n" ); document.write( "\n" ); document.write( "(6)*(-20)\r
\n" ); document.write( "\n" ); document.write( "(8)*(-15)\r
\n" ); document.write( "\n" ); document.write( "(10)*(-12)\r
\n" ); document.write( "\n" ); document.write( "(-1)*(120)\r
\n" ); document.write( "\n" ); document.write( "(-2)*(60)\r
\n" ); document.write( "\n" ); document.write( "(-3)*(40)\r
\n" ); document.write( "\n" ); document.write( "(-4)*(30)\r
\n" ); document.write( "\n" ); document.write( "(-5)*(24)\r
\n" ); document.write( "\n" ); document.write( "(-6)*(20)\r
\n" ); document.write( "\n" ); document.write( "(-8)*(15)\r
\n" ); document.write( "\n" ); document.write( "(-10)*(12)\r
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\n" ); document.write( "\n" ); document.write( "note: remember, the product of a negative and a positive number is a negative number\r
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\n" ); document.write( "\n" ); document.write( "Now which of these pairs add to -14? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -14\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 this list we can see that 6 and -20 add up to -14 and multiply to -120\r
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\n" ); document.write( "\n" ); document.write( "Now looking at the expression $\"8x%5E2-14x-15\"$ , replace $\"-14x\"$ with $\"6x%2B-20x\"$ (notice $\"6x%2B-20x\"$ adds up to $\"-14x\"$ . So it is equivalent to $\"-14x\"$ )\r
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\n" ); document.write( "\n" ); document.write( " $\"8x%5E2%2Bhighlight%286x%2B-20x%29%2B-15\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's factor $\"8x%5E2%2B6x-20x-15\"$ by grouping:\r
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\n" ); document.write( "\n" ); document.write( " $\"%288x%5E2%2B6x%29%2B%28-20x-15%29\"$ Group like terms\r
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\n" ); document.write( "\n" ); document.write( " $\"2x%284x%2B3%29-5%284x%2B3%29\"$ Factor out the GCF of $\"2x\"$ out of the first group. Factor out the GCF of $\"-5\"$ out of the second group\r
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\n" ); document.write( "\n" ); document.write( " $\"%282x-5%29%284x%2B3%29\"$ Since we have a common term of $\"4x%2B3\"$ , we can combine like terms\r
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\n" ); document.write( "\n" ); document.write( "So $\"8x%5E2%2B6x-20x-15\"$ factors to $\"%282x-5%29%284x%2B3%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So this also means that $\"8x%5E2-14x-15\"$ factors to $\"%282x-5%29%284x%2B3%29\"$ (since $\"8x%5E2-14x-15\"$ is equivalent to $\"8x%5E2%2B6x-20x-15\"$ )\r
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\n" ); document.write( "\n" ); document.write( "So our expression goes from $\"x%288x%5E2-14x-15%29\"$ and factors further to $\"x%282x-5%29%284x%2B3%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"8x%5E3-14x%5E2-15x\"$ factors to $\"x%282x-5%29%284x%2B3%29\"$ \r
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