document.write( "Question 1098636: Find a polynomial P(x) with real coefficients having a degree 6, leading coefficient 3, and zeros 6,0 (multiplicity 3), and 2-4i.
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Algebra.Com's Answer #713038 by greenestamps(13200) $\"\"$ $\"About$

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This is quite straightforward if you know the basic principles.

\n" ); document.write( "The root of 6 means the polynomial has a factor of (x-6); the root of 0 with multiplicity 3 means the polynomial has a factor of x (i.e., x-0) three times.

\n" ); document.write( "And since complex roots have to occur in conjugate pairs if the polynomial has real coefficients, the root of 2-4i means the polynomial has to have factors of (x-(2-4i)) and (x-(2+4i)).

\n" ); document.write( "And then there can be a scalar multiplier, a; so the polynomial is

\n" ); document.write( " $\"P%28x%29+=+a%28x%5E3%29%28x-6%29%28x-%282-4i%29%29%28x-%282%2B4i%29%29\"$

\n" ); document.write( "Presumably, if you are working a problem like this, you know how to expand that expression, if the answer is required in standard polynomial form. \n" ); document.write( "

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