document.write( "Question 215064: Completly factor the polynominal h^4+2h^3-8h^2\r
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Algebra.Com's Answer #162503 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!

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\n" ); document.write( "\n" ); document.write( " $\"h%5E4%2B2h%5E3-8h%5E2\"$ Start with the given expression.\r
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\n" ); document.write( "\n" ); document.write( " $\"h%5E2%28h%5E2%2B2h-8%29\"$ Factor out the GCF $\"h%5E2\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now let's try to factor the inner expression $\"h%5E2%2B2h-8\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"h%5E2%2B2h-8\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"2\"$ , and the last term is $\"-8\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"1\"$ by the last term $\"-8\"$ to get $\"%281%29%28-8%29=-8\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-8\"$ (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 $\"-8\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"-8\"$ :\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 $\"-8\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-8) = -8
\n" ); document.write( "2*(-4) = -8
\n" ); document.write( "(-1)*(8) = -8
\n" ); document.write( "(-2)*(4) = -8\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	-8	1+(-8)=-7
2	-4	2+(-4)=-2
-1	8	-1+8=7
-2	4	-2+4=2

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-2\"$ and $\"4\"$ add to $\"2\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-2\"$ and $\"4\"$ both multiply to $\"-8\"$ and add to $\"2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"2h\"$ with $\"-2h%2B4h\"$ . Remember, $\"-2\"$ and $\"4\"$ add to $\"2\"$ . So this shows us that $\"-2h%2B4h=2h\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"h%5E2%2Bhighlight%28-2h%2B4h%29-8\"$ Replace the second term $\"2h\"$ with $\"-2h%2B4h\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%28h%5E2-2h%29%2B%284h-8%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"h%28h-2%29%2B%284h-8%29\"$ Factor out the GCF $\"h\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"h%28h-2%29%2B4%28h-2%29\"$ 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( " $\"%28h%2B4%29%28h-2%29\"$ Combine like terms. Or factor out the common term $\"h-2\"$ \r
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\n" ); document.write( "\n" ); document.write( "So $\"h%5E2%28h%5E2%2B2h-8%29\"$ then factors further to $\"h%5E2%28h%2B4%29%28h-2%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"h%5E4%2B2h%5E3-8h%5E2\"$ completely factors to $\"h%5E2%28h%2B4%29%28h-2%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"h%5E4%2B2h%5E3-8h%5E2=h%5E2%28h%2B4%29%28h-2%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"h%5E2%28h%2B4%29%28h-2%29\"$ to get $\"h%5E4%2B2h%5E3-8h%5E2\"$ or by graphing the original expression and the answer (the two graphs should be identical).\r
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