document.write( "Question 1201674: Use the given zero to find the remaining zeros of the polynomial function \r
\n" ); document.write( "\n" ); document.write( "P(x) =x^4-6x^3+19x^2-6x+18; 3-3i \n" ); document.write( "
Algebra.Com's Answer #836158 by greenestamps(13200)\"\" \"About 
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\n" ); document.write( "The coefficients of the polynomial are real numbers, so the complex roots come in conjugate pairs. So if 3-3i is a root, 3+3i is another root.

\n" ); document.write( "You can use Vieta's Theorem to find the quadratic polynomial with roots 3-3i and 3+3i:
\n" ); document.write( "coefficient of linear term: -(sum of the two roots) = -((3-3i)+(3+3i)) = -6
\n" ); document.write( "constant term: product of the two roots: (3-3i)(3+3i) = 9-9i^3 = 9+9 = 18

\n" ); document.write( "The quadratic polynomial with roots 3-3i and 3+3i is x^2-6x+18. So

\n" ); document.write( "\"%28x%5E2-6x%2B18%29%28x%5E2%2Bax%2B1%29=x%5E4-6x%5E3%2B19x%5E2-6x%2B18\"

\n" ); document.write( "We know the leading coefficient of the second quadratic factor is 1, because the leading coefficient of the product is 1. And we know the constant term of the second quadratic factor is 1, because the constant term of the product is 18. So we find the constant \"a\" by looking at the linear term in the product.

\n" ); document.write( "On the left, the linear term is \"18a-6x\"; on the right it is \"-6x\". So a is 0; the other quadratic factor is \"x%5E2%2B1\". The zeros of that quadratic are i and -i.

\n" ); document.write( "ANSWER: given the one root 3-3i, the other roots are 3+3i, i, and -i.

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