document.write( "Question 977187: Hello!
\n" ); document.write( "What are all the rational zeros of the function?\r
\n" ); document.write( "\n" ); document.write( "f(x) = 2x^4 – 13x^3 + 12x^2 + 52x – 80\r
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Algebra.Com's Answer #598695 by Edwin McCravy(20054) $\"\"$ $\"About$

You can put this solution on YOUR website!
What are all the rational zeros of the function?\r
\n" ); document.write( "\n" ); document.write( "f(x) = 2x^4 – 13x^3 + 12x^2 + 52x – 80

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document.write( "It's in descending order, so look at the numbers on both ends,\r\n" );
document.write( "the 80 on the far right and the 2 on the far left.  Ignore signs\r\n" );
document.write( "for now. \r\n" );
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document.write( "All the factors of 80 are 1,2,4,5,8,10,16,20,40,80.\r\n" );
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document.write( "All the factors of 2 are 1 and 2.\r\n" );
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document.write( "Make all possible fractions with a numerator as a factor of 80 and\r\n" );
document.write( "a denominator as a factor of 2.  Some of the fractions will reduce\r\n" );
document.write( "and become duplicates.\r\n" );
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document.write( ",,,,,,,,,,,,,,,,,,,\r\n" );
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document.write( "Then we reduce the ones that will reduce and toss out the duplicates:\r\n" );
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document.write( ",,,,,,,,,,,,\r\n" );
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document.write( "Finally we put a  before every one because they could be either positive or\r\n" );
document.write( "negative.\r\n" );
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document.write( ",,,,,,,,,,,,\r\n" );
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document.write( "Now we have to use trial and error to find one of those, if there are any, that\r\n" );
document.write( "will prove to be a rational zero.  We try 1, the easiest one:\r\n" );
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document.write( "1 | 2 -13  12 52 -80\r\n" );
document.write( "  |     2 -11  1  53 \r\n" );
document.write( "    2 -11   1 53 -27\r\n" );
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document.write( "So 1 is not a zero, because we get -27 remainder instead of 0\r\n" );
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document.write( "So we try the next easiest one, 2\r\n" );
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document.write( "2 | 2 -13  12  52 -80\r\n" );
document.write( "  |     4 -18 -12  80 \r\n" );
document.write( "    2  -9  -6  40   0\r\n" );
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document.write( "So 2 is a zero, and now we have factored f(x) as\r\n" );
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document.write( "f(x) = (x-2)(2x³-9x²-6x+40) \r\n" );
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document.write( "Now we try the next easiest one, -1.  But we don't try it in the original\r\n" );
document.write( "polynomial.  Instead we try it in the cubic factor 2x³-9x²-6x+40 that we\r\n" );
document.write( "got from the synthetic division.\r\n" );
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document.write( "-1 | 2  -9  -6  40\r\n" );
document.write( "   |    -2  11  -5   \r\n" );
document.write( "     2 -11   5  35\r\n" );
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document.write( "So -1 is not a zero,\r\n" );
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document.write( "Now we try the next easiest one, -2\r\n" );
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document.write( "-2 | 2  -9  -6  40\r\n" );
document.write( "   |    -4  26 -40   \r\n" );
document.write( "     2 -13  20   0\r\n" );
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document.write( "So -2 is a zero, and now we have factored f(x) as\r\n" );
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document.write( "f(x) = (x-2)(x+2)(2x²-13x+20) \r\n" );
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document.write( "We can now use regular factor methods to factor 2x²-13x+20\r\n" );
document.write( "as (x-4)(x-5), and now we have factored the polynomial \r\n" );
document.write( "completely:\r\n" );
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document.write( "f(x) = (x-2)(x+2)(x-4)(2x-5)\r\n" );
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document.write( "The zeros all rational are found by setting each = 0 and solving.\r\n" );
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document.write( "x-2=0;  x+2=0;  x-4=0;  2x-5=0\r\n" );
document.write( "  x=2;    x=-2;   x=4     2x=5\r\n" );
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document.write( "Rational zeros: 2, -2, 4, and 5/2\r\n" );
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document.write( "Edwin

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