document.write( "Question 637203: We are supposed to factor polynomials and I just don't know what I'm doing!
\n" ); document.write( "3x^2-12x-63 \n" ); document.write( "

Algebra.Com's Answer #401496 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "\n" ); document.write( " $\"3x%5E2-12x-63\"$ Start with the given expression.\r
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\n" ); document.write( "\n" ); document.write( " $\"3%28x%5E2-4x-21%29\"$ Factor out the GCF $\"3\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now let's try to factor the inner expression $\"x%5E2-4x-21\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"x%5E2-4x-21\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"-4\"$ , and the last term is $\"-21\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"1\"$ by the last term $\"-21\"$ to get $\"%281%29%28-21%29=-21\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-21\"$ (the previous product) and add to the second coefficient $\"-4\"$ ?\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 $\"-21\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"-21\"$ :\r
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\n" ); document.write( "\n" ); document.write( "-1,-3,-7,-21\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 $\"-21\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-21) = -21
\n" ); document.write( "3*(-7) = -21
\n" ); document.write( "(-1)*(21) = -21
\n" ); document.write( "(-3)*(7) = -21\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 $\"-4\"$ :\r
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First Number	Second Number	Sum
1	-21	1+(-21)=-20
3	-7	3+(-7)=-4
-1	21	-1+21=20
-3	7	-3+7=4

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"3\"$ and $\"-7\"$ add to $\"-4\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"3\"$ and $\"-7\"$ both multiply to $\"-21\"$ and add to $\"-4\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-4x\"$ with $\"3x-7x\"$ . Remember, $\"3\"$ and $\"-7\"$ add to $\"-4\"$ . So this shows us that $\"3x-7x=-4x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"x%5E2%2Bhighlight%283x-7x%29-21\"$ Replace the second term $\"-4x\"$ with $\"3x-7x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%28x%5E2%2B3x%29%2B%28-7x-21%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%28x%2B3%29%2B%28-7x-21%29\"$ Factor out the GCF $\"x\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%28x%2B3%29-7%28x%2B3%29\"$ Factor out $\"7\"$ 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( " $\"%28x-7%29%28x%2B3%29\"$ Combine like terms. Or factor out the common term $\"x%2B3\"$ \r
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\n" ); document.write( "\n" ); document.write( "So $\"3%28x%5E2-4x-21%29\"$ then factors further to $\"3%28x-7%29%28x%2B3%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"3x%5E2-12x-63\"$ completely factors to $\"3%28x-7%29%28x%2B3%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"3x%5E2-12x-63=3%28x-7%29%28x%2B3%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"3%28x-7%29%28x%2B3%29\"$ to get $\"3x%5E2-12x-63\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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