document.write( "Question 550941: Factor the following expression completely:
\n" ); document.write( "70w^3 – 125w^2 + 30w \n" ); document.write( "

Algebra.Com's Answer #359240 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( " $\"70w%5E3-125w%5E2%2B30w\"$ Start with the given expression.\r
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\n" ); document.write( "\n" ); document.write( " $\"5w%2814w%5E2-25w%2B6%29\"$ Factor out the GCF $\"5w\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now let's try to factor the inner expression $\"14w%5E2-25w%2B6\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"14w%5E2-25w%2B6\"$ , we can see that the first coefficient is $\"14\"$ , the second coefficient is $\"-25\"$ , and the last term is $\"6\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"14\"$ by the last term $\"6\"$ to get $\"%2814%29%286%29=84\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"84\"$ (the previous product) and add to the second coefficient $\"-25\"$ ?\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 $\"84\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"84\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,6,7,12,14,21,28,42,84\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-6,-7,-12,-14,-21,-28,-42,-84\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 $\"84\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*84 = 84
\n" ); document.write( "2*42 = 84
\n" ); document.write( "3*28 = 84
\n" ); document.write( "4*21 = 84
\n" ); document.write( "6*14 = 84
\n" ); document.write( "7*12 = 84
\n" ); document.write( "(-1)*(-84) = 84
\n" ); document.write( "(-2)*(-42) = 84
\n" ); document.write( "(-3)*(-28) = 84
\n" ); document.write( "(-4)*(-21) = 84
\n" ); document.write( "(-6)*(-14) = 84
\n" ); document.write( "(-7)*(-12) = 84\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 $\"-25\"$ :\r
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First Number	Second Number	Sum
1	84	1+84=85
2	42	2+42=44
3	28	3+28=31
4	21	4+21=25
6	14	6+14=20
7	12	7+12=19
-1	-84	-1+(-84)=-85
-2	-42	-2+(-42)=-44
-3	-28	-3+(-28)=-31
-4	-21	-4+(-21)=-25
-6	-14	-6+(-14)=-20
-7	-12	-7+(-12)=-19

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-4\"$ and $\"-21\"$ add to $\"-25\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-4\"$ and $\"-21\"$ both multiply to $\"84\"$ and add to $\"-25\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-25w\"$ with $\"-4w-21w\"$ . Remember, $\"-4\"$ and $\"-21\"$ add to $\"-25\"$ . So this shows us that $\"-4w-21w=-25w\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"14w%5E2%2Bhighlight%28-4w-21w%29%2B6\"$ Replace the second term $\"-25w\"$ with $\"-4w-21w\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%2814w%5E2-4w%29%2B%28-21w%2B6%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"2w%287w-2%29%2B%28-21w%2B6%29\"$ Factor out the GCF $\"2w\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"2w%287w-2%29-3%287w-2%29\"$ Factor out $\"3\"$ 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( " $\"%282w-3%29%287w-2%29\"$ Combine like terms. Or factor out the common term $\"7w-2\"$ \r
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\n" ); document.write( "\n" ); document.write( "So $\"5w%2814w%5E2-25w%2B6%29\"$ then factors further to $\"5w%282w-3%29%287w-2%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"70w%5E3-125w%5E2%2B30w\"$ completely factors to $\"5w%282w-3%29%287w-2%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"70w%5E3-125w%5E2%2B30w=5w%282w-3%29%287w-2%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"5w%282w-3%29%287w-2%29\"$ to get $\"70w%5E3-125w%5E2%2B30w\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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