document.write( "Question 475599: Factor the polynomial completly\r
\n" ); document.write( "\n" ); document.write( "6x^2+11x-10 \n" ); document.write( "

Algebra.Com's Answer #326122 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"6x%5E2%2B11x-10\"$ , we can see that the first coefficient is $\"6\"$ , the second coefficient is $\"11\"$ , and the last term is $\"-10\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"6\"$ by the last term $\"-10\"$ to get $\"%286%29%28-10%29=-60\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-60\"$ (the previous product) and add to the second coefficient $\"11\"$ ?\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 $\"-60\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"-60\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,5,6,10,12,15,20,30,60\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-5,-6,-10,-12,-15,-20,-30,-60\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 $\"-60\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-60) = -60
\n" ); document.write( "2*(-30) = -60
\n" ); document.write( "3*(-20) = -60
\n" ); document.write( "4*(-15) = -60
\n" ); document.write( "5*(-12) = -60
\n" ); document.write( "6*(-10) = -60
\n" ); document.write( "(-1)*(60) = -60
\n" ); document.write( "(-2)*(30) = -60
\n" ); document.write( "(-3)*(20) = -60
\n" ); document.write( "(-4)*(15) = -60
\n" ); document.write( "(-5)*(12) = -60
\n" ); document.write( "(-6)*(10) = -60\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 $\"11\"$ :\r
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First Number	Second Number	Sum
1	-60	1+(-60)=-59
2	-30	2+(-30)=-28
3	-20	3+(-20)=-17
4	-15	4+(-15)=-11
5	-12	5+(-12)=-7
6	-10	6+(-10)=-4
-1	60	-1+60=59
-2	30	-2+30=28
-3	20	-3+20=17
-4	15	-4+15=11
-5	12	-5+12=7
-6	10	-6+10=4

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-4\"$ and $\"15\"$ add to $\"11\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-4\"$ and $\"15\"$ both multiply to $\"-60\"$ and add to $\"11\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"11x\"$ with $\"-4x%2B15x\"$ . Remember, $\"-4\"$ and $\"15\"$ add to $\"11\"$ . So this shows us that $\"-4x%2B15x=11x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"6x%5E2%2Bhighlight%28-4x%2B15x%29-10\"$ Replace the second term $\"11x\"$ with $\"-4x%2B15x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%286x%5E2-4x%29%2B%2815x-10%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"2x%283x-2%29%2B%2815x-10%29\"$ Factor out the GCF $\"2x\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"2x%283x-2%29%2B5%283x-2%29\"$ Factor out $\"5\"$ 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( " $\"%282x%2B5%29%283x-2%29\"$ Combine like terms. Or factor out the common term $\"3x-2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"6x%5E2%2B11x-10\"$ factors to $\"%282x%2B5%29%283x-2%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"6x%5E2%2B11x-10=%282x%2B5%29%283x-2%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%282x%2B5%29%283x-2%29\"$ to get $\"6x%5E2%2B11x-10\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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