document.write( "Question 111328: a^4-15a^2+56 \n" ); document.write( "

Algebra.Com's Answer #81231 by jim_thompson5910(35256) $\"\"$ $\"About$

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Looking at $\"a%5E4-15a%5E2%2B56\"$ we can see that the first term is $\"a%5E4\"$ and the last term is $\"56\"$ where the coefficients are 1 and 56 respectively.\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient 1 and the last coefficient 56 to get 56. Now what two numbers multiply to 56 and add to -15? Let's list all of the factors of 56:\r
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\n" ); document.write( "\n" ); document.write( "Factors of 56:\r
\n" ); document.write( "\n" ); document.write( "1,2,4,7,8,14,28,56\r
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\n" ); document.write( "\n" ); document.write( "-1,-2,-4,-7,-8,-14,-28,-56 ...List the negative factors as well. 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 56\r
\n" ); document.write( "\n" ); document.write( "1*56\r
\n" ); document.write( "\n" ); document.write( "2*28\r
\n" ); document.write( "\n" ); document.write( "4*14\r
\n" ); document.write( "\n" ); document.write( "7*8\r
\n" ); document.write( "\n" ); document.write( "(-1)*(-56)\r
\n" ); document.write( "\n" ); document.write( "(-2)*(-28)\r
\n" ); document.write( "\n" ); document.write( "(-4)*(-14)\r
\n" ); document.write( "\n" ); document.write( "(-7)*(-8)\r
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\n" ); document.write( "\n" ); document.write( "note: remember two negative numbers multiplied together make a positive number\r
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\n" ); document.write( "\n" ); document.write( "Now which of these pairs add to -15? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -15\r
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First Number	Second Number	Sum
1	56	1+56=57
2	28	2+28=30
4	14	4+14=18
7	8	7+8=15
-1	-56	-1+(-56)=-57
-2	-28	-2+(-28)=-30
-4	-14	-4+(-14)=-18
-7	-8	-7+(-8)=-15

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\n" ); document.write( "\n" ); document.write( "From this list we can see that -7 and -8 add up to -15 and multiply to 56\r
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\n" ); document.write( "\n" ); document.write( "Now looking at the expression $\"a%5E4-15a%5E2%2B56\"$ , replace $\"-15a%5E2\"$ with $\"-7a%5E2%2B-8a%5E2\"$ (notice $\"-7a%5E2%2B-8a%5E2\"$ adds up to $\"-15a%5E2\"$ . So it is equivalent to $\"-15a%5E2\"$ )\r
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\n" ); document.write( "\n" ); document.write( " $\"a%5E4%2Bhighlight%28-7a%5E2%2B-8a%5E2%29%2B56\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's factor $\"a%5E4-7a%5E2-8a%5E2%2B56\"$ by grouping:\r
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\n" ); document.write( "\n" ); document.write( " $\"%28a%5E4-7a%5E2%29%2B%28-8a%5E2%2B56%29\"$ Group like terms\r
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\n" ); document.write( "\n" ); document.write( " $\"a%5E2%28a%5E2-7%29-8%28a%5E2-7%29\"$ Factor out the GCF of $\"a%5E2\"$ out of the first group. Factor out the GCF of $\"8\"$ out of the second group\r
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\n" ); document.write( "\n" ); document.write( "So this also means that $\"a%5E4-15a%5E2%2B56\"$ factors to $\"%28a%5E2-8%29%28a%5E2-7%29\"$ (since $\"a%5E4-15a%5E2%2B56\"$ is equivalent to $\"a%5E4-7a%5E2-8a%5E2%2B56\"$ )
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