document.write( "Question 1132901: Using the given zero, find one other zero of f(x). Explain the process you used to find your solution.\r
\n" ); document.write( "\n" ); document.write( "1 - 6i is a zero of f(x) = x4 - 2x3 + 38x2 - 2x + 37. \n" ); document.write( "

Algebra.Com's Answer #750050 by greenestamps(13214) $\"\"$ $\"About$

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\n" ); document.write( "The polynomial has integer coefficients, and one of the roots is complex. That means another of the roots is the conjugate of the given complex number.

\n" ); document.write( "So another root is 1+6i; two of the roots are 1-6i and 1+6i.

\n" ); document.write( "The problem would have been more interesting -- and there would have been a lot more useful mathematics in it -- if it had asked us to find ALL the roots. So I'll go ahead and show you how that can be done.

\n" ); document.write( "Use Vieta's theorem to find the quadratic with those two roots:
\n" ); document.write( "(1) coefficient of the linear term is the opposite of the sum of the roots: -2
\n" ); document.write( "(2) constant term is the product of the roots: (1-6i)(1+6i) = 1+36 = 37

\n" ); document.write( "So the given polynomial is divisible by the quadratic polynomial x^2-2x+37. Long division (or any other method) then shows that the other quadratic factor is x^2+1; the roots of that quadratic are i and -i.

\n" ); document.write( "So the four roots of the given polynomial are i, -i, 1-6i, and 1+6i. \n" ); document.write( "

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