document.write( "Question 884723: Factoring Completrly: x^3+7x^2-x-7\r
\n" ); document.write( "\n" ); document.write( "The ac method: 3x^2+11x+10 \n" ); document.write( "

Algebra.Com's Answer #534610 by jim_thompson5910(35256) $\"\"$ $\"About$

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Problem #1: Factoring $\"x%5E3%2B7x%5E2-x-7\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"x%5E3%2B7x%5E2-x-7\"$ Start with the given expression.\r
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\n" ); document.write( "\n" ); document.write( " $\"%28x%5E3%2B7x%5E2%29%2B%28-x-7%29\"$ Group the terms\r
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\n" ); document.write( "\n" ); document.write( " $\"x%5E2%28x%2B7%29%2B%28-x-7%29\"$ Factor out $\"x%5E2\"$ from the first group\r
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\n" ); document.write( "\n" ); document.write( " $\"x%5E2%28x%2B7%29-1%28x%2B7%29\"$ Factor out $\"-1\"$ from the second group\r
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\n" ); document.write( "\n" ); document.write( " $\"%28x%5E2-1%29%28x%2B7%29\"$ Factor out $\"x%2B7\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"%28x-1%29%28x%2B1%29%28x%2B7%29\"$ Factor $\"x%5E2-1\"$ to get $\"%28x-1%29%28x%2B1%29\"$ (difference of squares rule)\r
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\n" ); document.write( "\n" ); document.write( "So $\"x%5E3%2B7x%5E2-x-7\"$ factors to $\"%28x-1%29%28x%2B1%29%28x%2B7%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "Problem # 2: Factoring $\"3x%5E2%2B11x%2B10\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"3x%5E2%2B11x%2B10\"$ , we can see that the first coefficient is $\"3\"$ , 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 $\"3\"$ by the last term $\"10\"$ to get $\"%283%29%2810%29=30\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"30\"$ (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 $\"30\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"30\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,5,6,10,15,30\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-5,-6,-10,-15,-30\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 $\"30\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*30 = 30
\n" ); document.write( "2*15 = 30
\n" ); document.write( "3*10 = 30
\n" ); document.write( "5*6 = 30
\n" ); document.write( "(-1)*(-30) = 30
\n" ); document.write( "(-2)*(-15) = 30
\n" ); document.write( "(-3)*(-10) = 30
\n" ); document.write( "(-5)*(-6) = 30\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	30	1+30=31
2	15	2+15=17
3	10	3+10=13
5	6	5+6=11
-1	-30	-1+(-30)=-31
-2	-15	-2+(-15)=-17
-3	-10	-3+(-10)=-13
-5	-6	-5+(-6)=-11

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"5\"$ and $\"6\"$ add to $\"11\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"5\"$ and $\"6\"$ both multiply to $\"30\"$ and add to $\"11\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"11x\"$ with $\"5x%2B6x\"$ . Remember, $\"5\"$ and $\"6\"$ add to $\"11\"$ . So this shows us that $\"5x%2B6x=11x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"3x%5E2%2Bhighlight%285x%2B6x%29%2B10\"$ Replace the second term $\"11x\"$ with $\"5x%2B6x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%283x%5E2%2B5x%29%2B%286x%2B10%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%283x%2B5%29%2B%286x%2B10%29\"$ Factor out the GCF $\"x\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%283x%2B5%29%2B2%283x%2B5%29\"$ Factor out $\"2\"$ 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( " $\"%28x%2B2%29%283x%2B5%29\"$ Combine like terms. Or factor out the common term $\"3x%2B5\"$ \r
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\n" ); document.write( "\n" ); document.write( "So $\"3x%5E2%2B11x%2B10\"$ factors to $\"%28x%2B2%29%283x%2B5%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"3x%5E2%2B11x%2B10=%28x%2B2%29%283x%2B5%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%28x%2B2%29%283x%2B5%29\"$ to get $\"3x%5E2%2B11x%2B10\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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