document.write( "Question 315391: Please help me factor this expression 36x^4-8x^3-28x^2 \n" ); document.write( "

Algebra.Com's Answer #225669 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( " $\"36x%5E4-8x%5E3-28x%5E2\"$ Start with the given expression.\r
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\n" ); document.write( "\n" ); document.write( " $\"4x%5E2%289x%5E2-2x-7%29\"$ Factor out the GCF $\"4x%5E2\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now let's try to factor the inner expression $\"9x%5E2-2x-7\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"9x%5E2-2x-7\"$ , we can see that the first coefficient is $\"9\"$ , the second coefficient is $\"-2\"$ , and the last term is $\"-7\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"9\"$ by the last term $\"-7\"$ to get $\"%289%29%28-7%29=-63\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-63\"$ (the previous product) and add to the second coefficient $\"-2\"$ ?\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 $\"-63\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"-63\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,3,7,9,21,63\r
\n" ); document.write( "\n" ); document.write( "-1,-3,-7,-9,-21,-63\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 $\"-63\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-63) = -63
\n" ); document.write( "3*(-21) = -63
\n" ); document.write( "7*(-9) = -63
\n" ); document.write( "(-1)*(63) = -63
\n" ); document.write( "(-3)*(21) = -63
\n" ); document.write( "(-7)*(9) = -63\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 $\"-2\"$ :\r
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First Number	Second Number	Sum
1	-63	1+(-63)=-62
3	-21	3+(-21)=-18
7	-9	7+(-9)=-2
-1	63	-1+63=62
-3	21	-3+21=18
-7	9	-7+9=2

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"7\"$ and $\"-9\"$ add to $\"-2\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"7\"$ and $\"-9\"$ both multiply to $\"-63\"$ and add to $\"-2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-2x\"$ with $\"7x-9x\"$ . Remember, $\"7\"$ and $\"-9\"$ add to $\"-2\"$ . So this shows us that $\"7x-9x=-2x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"9x%5E2%2Bhighlight%287x-9x%29-7\"$ Replace the second term $\"-2x\"$ with $\"7x-9x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%289x%5E2%2B7x%29%2B%28-9x-7%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%289x%2B7%29%2B%28-9x-7%29\"$ Factor out the GCF $\"x\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%289x%2B7%29-1%289x%2B7%29\"$ Factor out $\"1\"$ 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( " $\"%28x-1%29%289x%2B7%29\"$ Combine like terms. Or factor out the common term $\"9x%2B7\"$ \r
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\n" ); document.write( "\n" ); document.write( "So $\"4x%5E2%289x%5E2-2x-7%29\"$ then factors further to $\"4x%5E2%28x-1%29%289x%2B7%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"36x%5E4-8x%5E3-28x%5E2\"$ completely factors to $\"4x%5E2%28x-1%29%289x%2B7%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"36x%5E4-8x%5E3-28x%5E2=4x%5E2%28x-1%29%289x%2B7%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"4x%5E2%28x-1%29%289x%2B7%29\"$ to get $\"36x%5E4-8x%5E3-28x%5E2\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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