document.write( "Question 1072882: Find a fourth degree polynomial function f(x) with real coefficients that has i and 3i as zeros such that f(-1)=20 \n" ); document.write( "

Algebra.Com's Answer #687730 by KMST(5328) $\"\"$ $\"About$

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If a polynomial has real coefficients,
\n" ); document.write( "and a complex zero,
\n" ); document.write( "the conjugate complex number should also be a zero.
\n" ); document.write( "In other words, if $\"a%2Bb%2Ai\"$ is a zero, so is $\"a-b%2Ai\"$ .
\n" ); document.write( "In this case, $\"-i\"$ and $\"-3i\"$ are also zeros.
\n" ); document.write( "With those four different zeros, a fourth degree polynomial must have the factored form
\n" ); document.write( " $\"f%28x%29=c%28x-i%29%28x-%28-i%29%29%28x-3i%29%28x-%28-3i%29%29\"$ for some real non-zero number $\"c\"$ .
\n" ); document.write( "Simplifying,
\n" ); document.write( " $\"f%28x%29=c%28x-i%29%28x%2Bi%29%28x-3i%29%28x%2B3i%29\"$
\n" ); document.write( " $\"f%28x%29=c%28x%5E2-i%5E2%29%28x%5E2-%283i%29%5E2%29\"$
\n" ); document.write( " $\"f%28x%29=c%28x%5E2-%28-1%29%29%28x%5E2-3%5E2i%5E2%29\"$
\n" ); document.write( " $\"f%28x%29=c%28x%5E2%2B1%29%28x%5E2-9%28-1%29%29\"$
\n" ); document.write( " $\"f%28x%29=c%28x%5E2%2B1%29%28x%5E2%2B9%29\"$
\n" ); document.write( " $\"system%28f%28-1%29=20%2Cf%28x%29=c%28x%5E2%2B1%29%28x%5E2%2B9%29%29\"$ -->

--> $\"system%28c%281%2B1%29%281%2B9%29=20%2Cf%28x%29=c%28x%5E2%2B1%29%28x%5E2%2B9%29%29\"$ --> $\"system%28c%2A2%2A10=20%2Cf%28x%29=c%28x%5E2%2B1%29%28x%5E2%2B9%29%29\"$ --> $\"system%28c=1%2Cf%28x%29=%28x%5E2%2B1%29%28x%5E2%2B9%29%29\"$ .
\n" ); document.write( "So $\"highlight%28f%28x%29=%28x%5E2%2B1%29%28x%5E2%2B9%29%29\"$ or multiplying the factors $\"highlight%28f%28x%29=x%5E4%2B10x%5E2%2B9%29\"$ . \n" ); document.write( "

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