document.write( "Question 238361: Please help me find the real zeros of this polynomial function:\r
\n" ); document.write( "\n" ); document.write( " $\"3x%5E4-8x%5E3-5x%5E2%2B16x-5=f%28x%29\"$ \r
\n" ); document.write( "\n" ); document.write( "Please find the zeros using the Rational Zero Theorem and Synthetic Division.\r
\n" ); document.write( "\n" ); document.write( "Thank you! \n" ); document.write( "

Algebra.Com's Answer #175160 by jsmallt9(3758) $\"\"$ $\"About$

You can put this solution on YOUR website!
The possible rational roots are all the rational numbers, positive and negative, which can be formed using a factor of the constant term (at the end) over a factor of the leading coefficient.

\n" ); document.write( "This function's constant term is -5 and its leading coefficient if 3. These are both prime numbers so the number of possible rational roots is small: 1/1, 5/1, 1/3, and 5/3 (both positive and negative).

\n" ); document.write( "Synthetic Division is a relatively quick and easy way to determine which, if any of these possible roots actually are roots. Here's how to test to see if 1/1 (aka 1):
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document.write( "1 |   3   -8   -5   16   -5\r\n" );
document.write( "---        3   -5  -10    6\r\n" );
document.write( "     -----------------------\r\n" );
document.write( "      3   -5  -10    6    1\r\n" );
document.write( "Since the remainder (the \"1\" in the lower right above) is not 0, 1 is not a root for f(x).\r\n" );
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\n" ); document.write( "There is a lot of trial and error in this process. (Don't forget to try the possible negative rational roots, even if both the constant term and the leading coefficient are positive.) I have tried the possible rational roots and I was not able to find any. This means one of the following:

There are no rational roots. This is possible. But teachers don't usually ask you to find roots when they can't be found.
There is an error in the function as you've written it. Please double check the problem and, if there was an error, resubmit a corrected function.
I have made a calculation error when I tried the various possible roots. This is possible, even though I checked my work. So if the function you have given here is correct, I would go ahead and try to find them on your own.

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