document.write( "Question 390785: Make up a polynomial that meets the following conditions:
\n" ); document.write( "a)it has rational zeros 3/4 and -1/3
\n" ); document.write( "b)it has two complex roots of the form a+bi,a,b \"not equal\" to 0
\n" ); document.write( "Then, create the list of all possible rational roots and use synthetic division to reduce the 4th degree polynomial to a quadratic and solve for the remaining roots. Show all steps. \n" ); document.write( "

Algebra.Com's Answer #277147 by richard1234(7193) $\"\"$ $\"About$

You can put this solution on YOUR website!
There's an important theorem called \"The Fundamental Theorem of Algebra\" (it looks easy, but requires high-level analysis to prove) that says that any n-degree polynomial has n roots, counting multiplicities.\r
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\n" ); document.write( "\n" ); document.write( "Furthermore, if r is a root of the polynomial, then (x-r) is a factor of it (this will be useful in generating the polynomial).\r
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\n" ); document.write( "\n" ); document.write( "Therefore the polynomial P(x) is equal to\r
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\n" ); document.write( "\n" ); document.write( " $\"P%28x%29+=+%28x-3%2F4%29%28x%2B1%2F3%29%28x+-+a+-+bi%29%28x+-+a+-+bi%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "You write in your problem \"Then, create the list of all possible rational roots and use synthetic division to reduce the 4th degree polynomial to a quadratic and solve for the remaining roots. Show all steps.\" However, the only rational roots are 3/4 and -1/3. The quadratic with a double root x - a - bi has double root $\"a+%2B+bi\"$ . Therefore, solving for the remaining roots using synthetic division is unnecessary...only do that if you're practicing your synthetic division :) \n" ); document.write( "

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