document.write( "Question 115752: Factor these ?\r
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Algebra.Com's Answer #84217 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
#1\r
\n" ); document.write( "\n" ); document.write( "Looking at $\"4x%5E2-12x-40\"$ we can see that the first term is $\"4x%5E2\"$ and the last term is $\"-40\"$ where the coefficients are 4 and -40 respectively.\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient 4 and the last coefficient -40 to get -160. Now what two numbers multiply to -160 and add to the middle coefficient -12? Let's list all of the factors of -160:\r
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\n" ); document.write( "\n" ); document.write( "Factors of -160:\r
\n" ); document.write( "\n" ); document.write( "1,2,4,5,8,10,16,20,32,40,80,160\r
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\n" ); document.write( "\n" ); document.write( "-1,-2,-4,-5,-8,-10,-16,-20,-32,-40,-80,-160 ...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 -160\r
\n" ); document.write( "\n" ); document.write( "(1)*(-160)\r
\n" ); document.write( "\n" ); document.write( "(2)*(-80)\r
\n" ); document.write( "\n" ); document.write( "(4)*(-40)\r
\n" ); document.write( "\n" ); document.write( "(5)*(-32)\r
\n" ); document.write( "\n" ); document.write( "(8)*(-20)\r
\n" ); document.write( "\n" ); document.write( "(10)*(-16)\r
\n" ); document.write( "\n" ); document.write( "(-1)*(160)\r
\n" ); document.write( "\n" ); document.write( "(-2)*(80)\r
\n" ); document.write( "\n" ); document.write( "(-4)*(40)\r
\n" ); document.write( "\n" ); document.write( "(-5)*(32)\r
\n" ); document.write( "\n" ); document.write( "(-8)*(20)\r
\n" ); document.write( "\n" ); document.write( "(-10)*(16)\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 -12? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -12\r
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First Number	Second Number	Sum
1	-160	1+(-160)=-159
2	-80	2+(-80)=-78
4	-40	4+(-40)=-36
5	-32	5+(-32)=-27
8	-20	8+(-20)=-12
10	-16	10+(-16)=-6
-1	160	-1+160=159
-2	80	-2+80=78
-4	40	-4+40=36
-5	32	-5+32=27
-8	20	-8+20=12
-10	16	-10+16=6

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\n" ); document.write( "\n" ); document.write( "From this list we can see that 8 and -20 add up to -12 and multiply to -160\r
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\n" ); document.write( "\n" ); document.write( "Now looking at the expression $\"4x%5E2-12x-40\"$ , replace $\"-12x\"$ with $\"8x%2B-20x\"$ (notice $\"8x%2B-20x\"$ adds up to $\"-12x\"$ . So it is equivalent to $\"-12x\"$ )\r
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\n" ); document.write( "\n" ); document.write( " $\"4x%5E2%2Bhighlight%288x%2B-20x%29%2B-40\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's factor $\"4x%5E2%2B8x-20x-40\"$ by grouping:\r
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\n" ); document.write( "\n" ); document.write( " $\"%284x%5E2%2B8x%29%2B%28-20x-40%29\"$ Group like terms\r
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\n" ); document.write( "\n" ); document.write( " $\"4x%28x%2B2%29-20%28x%2B2%29\"$ Factor out the GCF of $\"4x\"$ out of the first group. Factor out the GCF of $\"-20\"$ out of the second group\r
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\n" ); document.write( "\n" ); document.write( " $\"%284x-20%29%28x%2B2%29\"$ Since we have a common term of $\"x%2B2\"$ , we can combine like terms\r
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\n" ); document.write( "\n" ); document.write( "So $\"4x%5E2%2B8x-20x-40\"$ factors to $\"%284x-20%29%28x%2B2%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So this also means that $\"4x%5E2-12x-40\"$ factors to $\"%284x-20%29%28x%2B2%29\"$ (since $\"4x%5E2-12x-40\"$ is equivalent to $\"4x%5E2%2B8x-20x-40\"$ )\r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"4x%5E2-12x-40\"$ factors to $\"%284x-20%29%28x%2B2%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at $\"2x%5E2-16x%2B30\"$ we can see that the first term is $\"2x%5E2\"$ and the last term is $\"30\"$ where the coefficients are 2 and 30 respectively.\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient 2 and the last coefficient 30 to get 60. Now what two numbers multiply to 60 and add to the middle coefficient -16? Let's list all of the factors of 60:\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
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\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-5,-6,-10,-12,-15,-20,-30,-60 ...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 60\r
\n" ); document.write( "\n" ); document.write( "1*60\r
\n" ); document.write( "\n" ); document.write( "2*30\r
\n" ); document.write( "\n" ); document.write( "3*20\r
\n" ); document.write( "\n" ); document.write( "4*15\r
\n" ); document.write( "\n" ); document.write( "5*12\r
\n" ); document.write( "\n" ); document.write( "6*10\r
\n" ); document.write( "\n" ); document.write( "(-1)*(-60)\r
\n" ); document.write( "\n" ); document.write( "(-2)*(-30)\r
\n" ); document.write( "\n" ); document.write( "(-3)*(-20)\r
\n" ); document.write( "\n" ); document.write( "(-4)*(-15)\r
\n" ); document.write( "\n" ); document.write( "(-5)*(-12)\r
\n" ); document.write( "\n" ); document.write( "(-6)*(-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 -16? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -16\r
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First Number	Second Number	Sum
1	60	1+60=61
2	30	2+30=32
3	20	3+20=23
4	15	4+15=19
5	12	5+12=17
6	10	6+10=16
-1	-60	-1+(-60)=-61
-2	-30	-2+(-30)=-32
-3	-20	-3+(-20)=-23
-4	-15	-4+(-15)=-19
-5	-12	-5+(-12)=-17
-6	-10	-6+(-10)=-16

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\n" ); document.write( "\n" ); document.write( "From this list we can see that -6 and -10 add up to -16 and multiply to 60\r
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\n" ); document.write( "\n" ); document.write( "Now looking at the expression $\"2x%5E2-16x%2B30\"$ , replace $\"-16x\"$ with $\"-6x%2B-10x\"$ (notice $\"-6x%2B-10x\"$ adds up to $\"-16x\"$ . So it is equivalent to $\"-16x\"$ )\r
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\n" ); document.write( "\n" ); document.write( " $\"2x%5E2%2Bhighlight%28-6x%2B-10x%29%2B30\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's factor $\"2x%5E2-6x-10x%2B30\"$ by grouping:\r
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\n" ); document.write( "\n" ); document.write( " $\"%282x%5E2-6x%29%2B%28-10x%2B30%29\"$ Group like terms\r
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\n" ); document.write( "\n" ); document.write( " $\"2x%28x-3%29-10%28x-3%29\"$ Factor out the GCF of $\"2x\"$ out of the first group. Factor out the GCF of $\"-10\"$ out of the second group\r
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\n" ); document.write( "\n" ); document.write( " $\"%282x-10%29%28x-3%29\"$ Since we have a common term of $\"x-3\"$ , we can combine like terms\r
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\n" ); document.write( "\n" ); document.write( "So $\"2x%5E2-6x-10x%2B30\"$ factors to $\"%282x-10%29%28x-3%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So this also means that $\"2x%5E2-16x%2B30\"$ factors to $\"%282x-10%29%28x-3%29\"$ (since $\"2x%5E2-16x%2B30\"$ is equivalent to $\"2x%5E2-6x-10x%2B30\"$ )\r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"2x%5E2-16x%2B30\"$ factors to $\"%282x-10%29%28x-3%29\"$ \r
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