document.write( "Question 1204260: List all possible rational zeros given by the Rational Zeros Theorem (but don't check to see which actually are zeros). (Enter your answers as a comma-separated list.)
\n" ); document.write( "U(x) = 12x^5 + 7x^3 − 3x − 8
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\n" ); document.write( "Leading coefficient = 12, which is always attached to the term with the largest exponent
\n" ); document.write( "Constant term = -8\r
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\n" ); document.write( "\n" ); document.write( "Positive factors of the constant term: 1, 2, 4, 8
\n" ); document.write( "Positive factors of the leading coefficient: 1, 2, 3, 4, 6, 12\r
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\n" ); document.write( "\n" ); document.write( "Each possible rational root will be of the form p/q
\n" ); document.write( "p = factor of the constant term
\n" ); document.write( "q = factor of the leading term\r
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\n" ); document.write( "\n" ); document.write( "The way I like to do this kind of problem is to arrange the factors mentioned in a table like this
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	1	2	4	8
1
2
3
4
6
12

\r
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\n" ); document.write( "\n" ); document.write( "Then to fill out the first row, we divide the items in the upper row header (1,2,4 and 8) by the value 1.
\n" ); document.write( "The values won't change.
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	1	2	4	8
1	1	2	4	8
2
3
4
6
12

\n" ); document.write( "The second row is a bit more interesting. We divide the items in the upper row (1,2,4,8) by the value 2. Each value in the second row is fully reduced. Eg: 2/2 reduces to 1.
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	1	2	4	8
1	1	2	4	8
2	1/2	1	2	4
3
4
6
12

\n" ); document.write( "This process is carried out to have this completed table.
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	1	2	4	8
1	1	2	4	8
2	1/2	1	2	4
3	1/3	2/3	4/3	8/3
4	1/4	1/2	1	2
6	1/6	1/3	2/3	4/3
12	1/12	1/6	1/3	2/3

\n" ); document.write( "At first glance it appears there are 4*6 = 24 possible positive rational roots. \r
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\n" ); document.write( "\n" ); document.write( "However, there are a few repeated items. For instance, \"2/3\" shows up 3 times.
\n" ); document.write( "After removing duplicates, we have these possible positive rational roots:
\n" ); document.write( "1, 2, 4, 8, 1/2, 1/3, 2/3, 4/3, 8/3, 1/4, 1/6, 1/12\r
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\n" ); document.write( "\n" ); document.write( "List the plus and minus version of each root to get the full set of possible rational roots.
\n" ); document.write( "Of course, U(x) will only have up to 5 rational roots since this is the degree of the polynomial (it's the largest exponent).\r
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\n" ); document.write( "\n" ); document.write( "Answer:
\n" ); document.write( "±1, ±2, ±4, ±8, ±1/2, ±1/3, ±2/3, ±4/3, ±8/3, ±1/4, ±1/6, ±1/12\r
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\n" ); document.write( "\n" ); document.write( "Feel free to find other ways to arrange the values to what makes the most sense for you.
\n" ); document.write( "The way I've arranged them is to put the whole numbers first, then fractions afterward.
\n" ); document.write( "The denominators are grouped together. Within the subgroup that has denominator 3, the numerators increase from left to right.\r
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\n" ); document.write( "\n" ); document.write( "A lengthier way to write the list would be to say:
\n" ); document.write( "1, -1, 2, -2, 4, -4, 8, -8, 1/2, -1/2, 1/3, -1/3, 2/3, -2/3, 4/3, -4/3, 8/3, -8/3, 1/4, -1/4, 1/6, -1/6, 1/12, -1/12
\n" ); document.write( "The first method is more efficient since we don't have to write the same value twice more or less.
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