document.write( "Question 251559: 64a^4-224a^2+196= \n" ); document.write( "

Algebra.Com's Answer #183242 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( " $\"64a%5E4-224a%5E2%2B196\"$ Start with the given expression\r
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\n" ); document.write( "\n" ); document.write( " $\"4%2816a%5E4-56a%5E2%2B49%29\"$ Factor out the GCF $\"4\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's focus on the inner expression $\"16a%5E4-56a%5E2%2B49\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at $\"16a%5E4-56a%5E2%2B49\"$ we can see that the first term is $\"16a%5E4\"$ and the last term is $\"49\"$ where the coefficients are 16 and 49 respectively.\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient 16 and the last coefficient 49 to get 784. Now what two numbers multiply to 784 and add to the middle coefficient -56? Let's list all of the factors of 784:\r
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\n" ); document.write( "\n" ); document.write( "Factors of 784:\r
\n" ); document.write( "\n" ); document.write( "1,2,4,7,8,14,16,28,49,56,98,112,196,392\r
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\n" ); document.write( "\n" ); document.write( "-1,-2,-4,-7,-8,-14,-16,-28,-49,-56,-98,-112,-196,-392 ...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 784\r
\n" ); document.write( "\n" ); document.write( "1*784\r
\n" ); document.write( "\n" ); document.write( "2*392\r
\n" ); document.write( "\n" ); document.write( "4*196\r
\n" ); document.write( "\n" ); document.write( "7*112\r
\n" ); document.write( "\n" ); document.write( "8*98\r
\n" ); document.write( "\n" ); document.write( "14*56\r
\n" ); document.write( "\n" ); document.write( "16*49\r
\n" ); document.write( "\n" ); document.write( "28*28\r
\n" ); document.write( "\n" ); document.write( "(-1)*(-784)\r
\n" ); document.write( "\n" ); document.write( "(-2)*(-392)\r
\n" ); document.write( "\n" ); document.write( "(-4)*(-196)\r
\n" ); document.write( "\n" ); document.write( "(-7)*(-112)\r
\n" ); document.write( "\n" ); document.write( "(-8)*(-98)\r
\n" ); document.write( "\n" ); document.write( "(-14)*(-56)\r
\n" ); document.write( "\n" ); document.write( "(-16)*(-49)\r
\n" ); document.write( "\n" ); document.write( "(-28)*(-28)\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 -56? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -56\r
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First Number	Second Number	Sum
1	784	1+784=785
2	392	2+392=394
4	196	4+196=200
7	112	7+112=119
8	98	8+98=106
14	56	14+56=70
16	49	16+49=65
28	28	28+28=56
-1	-784	-1+(-784)=-785
-2	-392	-2+(-392)=-394
-4	-196	-4+(-196)=-200
-7	-112	-7+(-112)=-119
-8	-98	-8+(-98)=-106
-14	-56	-14+(-56)=-70
-16	-49	-16+(-49)=-65
-28	-28	-28+(-28)=-56

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\n" ); document.write( "\n" ); document.write( "From this list we can see that -28 and -28 add up to -56 and multiply to 784\r
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\n" ); document.write( "\n" ); document.write( "Now looking at the expression $\"16a%5E4-56a%5E2%2B49\"$ , replace $\"-56a%5E2\"$ with $\"-28a%5E2-28a%5E2\"$ (notice $\"-28a%5E2-28a%5E2\"$ combines back to $\"-56a%5E2\"$ . So it is equivalent to $\"-56a%5E2\"$ )\r
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\n" ); document.write( "\n" ); document.write( " $\"16a%5E4%2Bhighlight%28-28a%5E2-28a%5E2%29%2B49\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's factor $\"16a%5E4-28a%5E2-28a%5E2%2B49\"$ by grouping:\r
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\n" ); document.write( "\n" ); document.write( " $\"%2816a%5E4-28a%5E2%29%2B%28-28a%5E2%2B49%29\"$ Group like terms\r
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\n" ); document.write( "\n" ); document.write( " $\"4a%5E2%284a%5E2-7%29-7%284a%5E2-7%29\"$ Factor out the GCF of $\"4a%5E2\"$ out of the first group. Factor out the GCF of $\"-7\"$ out of the second group\r
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\n" ); document.write( "\n" ); document.write( " $\"%284a%5E2-7%29%284a%5E2-7%29\"$ Since we have a common term of $\"4a%5E2-7\"$ , we can combine like terms\r
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\n" ); document.write( "\n" ); document.write( "So $\"16a%5E4-28a%5E2-28a%5E2%2B49\"$ factors to $\"%284a%5E2-7%29%284a%5E2-7%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So this also means that $\"16a%5E4-56a%5E2%2B49\"$ factors to $\"%284a%5E2-7%29%284a%5E2-7%29\"$ (since $\"16a%5E4-56a%5E2%2B49\"$ is equivalent to $\"16a%5E4-28a%5E2-28a%5E2%2B49\"$ )\r
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\n" ); document.write( "\n" ); document.write( "note: $\"%284a%5E2-7%29%284a%5E2-7%29\"$ is equivalent to $\"%284a%5E2-7%29%5E2\"$ since the term $\"4a%5E2-7\"$ occurs twice. So $\"16a%5E4-56a%5E2%2B49\"$ also factors to $\"%284a%5E2-7%29%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "So our expression goes from $\"4%2816a%5E4-56a%5E2%2B49%29\"$ and factors further to $\"4%284a%5E2-7%29%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "So $\"64a%5E4-224a%5E2%2B196\"$ factors to $\"4%284a%5E2-7%29%5E2\"$ \r
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