document.write( "Question 282613: factor 3w^2 - 8w -16 \n" ); document.write( "

Algebra.Com's Answer #205202 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"3w%5E2-8w-16\"$ , we can see that the first coefficient is $\"3\"$ , the second coefficient is $\"-8\"$ , and the last term is $\"-16\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"3\"$ by the last term $\"-16\"$ to get $\"%283%29%28-16%29=-48\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-48\"$ (the previous product) and add to the second coefficient $\"-8\"$ ?\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 $\"-48\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"-48\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,6,8,12,16,24,48\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-6,-8,-12,-16,-24,-48\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 $\"-48\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-48) = -48
\n" ); document.write( "2*(-24) = -48
\n" ); document.write( "3*(-16) = -48
\n" ); document.write( "4*(-12) = -48
\n" ); document.write( "6*(-8) = -48
\n" ); document.write( "(-1)*(48) = -48
\n" ); document.write( "(-2)*(24) = -48
\n" ); document.write( "(-3)*(16) = -48
\n" ); document.write( "(-4)*(12) = -48
\n" ); document.write( "(-6)*(8) = -48\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 $\"-8\"$ :\r
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First Number	Second Number	Sum
1	-48	1+(-48)=-47
2	-24	2+(-24)=-22
3	-16	3+(-16)=-13
4	-12	4+(-12)=-8
6	-8	6+(-8)=-2
-1	48	-1+48=47
-2	24	-2+24=22
-3	16	-3+16=13
-4	12	-4+12=8
-6	8	-6+8=2

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"4\"$ and $\"-12\"$ add to $\"-8\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"4\"$ and $\"-12\"$ both multiply to $\"-48\"$ and add to $\"-8\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-8w\"$ with $\"4w-12w\"$ . Remember, $\"4\"$ and $\"-12\"$ add to $\"-8\"$ . So this shows us that $\"4w-12w=-8w\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"3w%5E2%2Bhighlight%284w-12w%29-16\"$ Replace the second term $\"-8w\"$ with $\"4w-12w\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%283w%5E2%2B4w%29%2B%28-12w-16%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"w%283w%2B4%29%2B%28-12w-16%29\"$ Factor out the GCF $\"w\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"w%283w%2B4%29-4%283w%2B4%29\"$ Factor out $\"4\"$ 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( " $\"%28w-4%29%283w%2B4%29\"$ Combine like terms. Or factor out the common term $\"3w%2B4\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"3w%5E2-8w-16\"$ factors to $\"%28w-4%29%283w%2B4%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"3w%5E2-8w-16=%28w-4%29%283w%2B4%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%28w-4%29%283w%2B4%29\"$ to get $\"3w%5E2-8w-16\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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