document.write( "Question 571790: find a third-degree polynomial equation with rational coefficients that has 1 and 3i as roots. \n" ); document.write( "

Algebra.Com's Answer #368194 by KMST(5328) $\"\"$ $\"About$

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If the coefficients are rational numbers, irrational roots have to appear in conjugate pairs, so -3i must also be a root.
\n" ); document.write( "A polynomial of degree 3, with 1, 3i, and -3i as roots has to be equal to
\n" ); document.write( " $\"K%28x-1%29%28x-3i%29%28x%2B3i%29\"$ with some rational non-zero number $\"K\"$ , and all polynomials of such form are of degree 3, and have 1, 3i, and -3i as roots.
\n" ); document.write( "So there are infinite such polynomials for the answer, but the simplest, with $\"K=1\"$ is
\n" ); document.write( " $\"%28x-1%29%28x-3i%29%28x%2B3i%29=%28x-1%29%28x%5E2%2B9%29=x%5E3-x%5E2%2B9x-9\"$ \n" ); document.write( "

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