document.write( "Question 1000695: 5+i and 1+4i
\n" ); document.write( "Find a polynomial function f(x) of least degree having only real coefficients with zeros as given \n" ); document.write( "

Algebra.Com's Answer #618039 by ikleyn(52799) $\"\"$ $\"About$

You can put this solution on YOUR website!
.
\n" ); document.write( "If a polynomial with real coefficients has the zero 5+i, then it has 5-i as a zero too.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "If a polynomial with real coefficients has the zero 1+4i, then it has 1-4i as a zero too.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Thus the polynomial of the minimal degree with real coefficients is\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "p(x) = (x-(5+i))*(x-(5-i))*(x-(1+4i))*(x-(1-4i)).\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now, (x-(5+i))*(x-(5-i)) = $\"x%5E2+-+10x+%2B+26\"$ ,\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "(x-(1+4i))*(x-(1-4i)) = $\"x%5E2+-+2x+%2B+17\"$ .\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Hence,\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "p(x) = $\"%28x%5E2+-+10x+%2B+26%29\"$ . $\"%28x%5E2+-+2x+%2B+17%29\"$ .\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "You can open parentheses if you want/you need.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

\n" );