document.write( "Question 117273: Factor completely. 15x^2 – 21x + 6 \n" ); document.write( "

Algebra.Com's Answer #85332 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at $\"15x%5E2-21x%2B6\"$ we can see that the first term is $\"15x%5E2\"$ and the last term is $\"6\"$ where the coefficients are 15 and 6 respectively.\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient 15 and the last coefficient 6 to get 90. Now what two numbers multiply to 90 and add to the middle coefficient -21? Let's list all of the factors of 90:\r
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\n" ); document.write( "\n" ); document.write( "Factors of 90:\r
\n" ); document.write( "\n" ); document.write( "1,2,3,5,6,9,10,15,18,30,45,90\r
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\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-5,-6,-9,-10,-15,-18,-30,-45,-90 ...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 90\r
\n" ); document.write( "\n" ); document.write( "1*90\r
\n" ); document.write( "\n" ); document.write( "2*45\r
\n" ); document.write( "\n" ); document.write( "3*30\r
\n" ); document.write( "\n" ); document.write( "5*18\r
\n" ); document.write( "\n" ); document.write( "6*15\r
\n" ); document.write( "\n" ); document.write( "9*10\r
\n" ); document.write( "\n" ); document.write( "(-1)*(-90)\r
\n" ); document.write( "\n" ); document.write( "(-2)*(-45)\r
\n" ); document.write( "\n" ); document.write( "(-3)*(-30)\r
\n" ); document.write( "\n" ); document.write( "(-5)*(-18)\r
\n" ); document.write( "\n" ); document.write( "(-6)*(-15)\r
\n" ); document.write( "\n" ); document.write( "(-9)*(-10)\r
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\n" ); document.write( "\n" ); document.write( "note: remember two negative numbers multiplied together make a positive number\r
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\n" ); document.write( "\n" ); document.write( "Now which of these pairs add to -21? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -21\r
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First Number	Second Number	Sum
1	90	1+90=91
2	45	2+45=47
3	30	3+30=33
5	18	5+18=23
6	15	6+15=21
9	10	9+10=19
-1	-90	-1+(-90)=-91
-2	-45	-2+(-45)=-47
-3	-30	-3+(-30)=-33
-5	-18	-5+(-18)=-23
-6	-15	-6+(-15)=-21
-9	-10	-9+(-10)=-19

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\n" ); document.write( "\n" ); document.write( "From this list we can see that -6 and -15 add up to -21 and multiply to 90\r
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\n" ); document.write( "\n" ); document.write( "Now looking at the expression $\"15x%5E2-21x%2B6\"$ , replace $\"-21x\"$ with $\"-6x%2B-15x\"$ (notice $\"-6x%2B-15x\"$ adds up to $\"-21x\"$ . So it is equivalent to $\"-21x\"$ )\r
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\n" ); document.write( "\n" ); document.write( " $\"15x%5E2%2Bhighlight%28-6x%2B-15x%29%2B6\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's factor $\"15x%5E2-6x-15x%2B6\"$ by grouping:\r
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\n" ); document.write( "\n" ); document.write( " $\"%2815x%5E2-6x%29%2B%28-15x%2B6%29\"$ Group like terms\r
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\n" ); document.write( "\n" ); document.write( " $\"3x%285x-2%29-3%285x-2%29\"$ Factor out the GCF of $\"3x\"$ out of the first group. Factor out the GCF of $\"-3\"$ out of the second group\r
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\n" ); document.write( "\n" ); document.write( " $\"%283x-3%29%285x-2%29\"$ Since we have a common term of $\"5x-2\"$ , we can combine like terms\r
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\n" ); document.write( "\n" ); document.write( "So $\"15x%5E2-6x-15x%2B6\"$ factors to $\"%283x-3%29%285x-2%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So this also means that $\"15x%5E2-21x%2B6\"$ factors to $\"%283x-3%29%285x-2%29\"$ (since $\"15x%5E2-21x%2B6\"$ is equivalent to $\"15x%5E2-6x-15x%2B6\"$ )\r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"15x%5E2-21x%2B6\"$ factors to $\"%283x-3%29%285x-2%29\"$ \n" ); document.write( "

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