document.write( "Question 977119: find the root of the polynomial p(x)=x^4 + 4x^3 +6x^2 + 4x + 5 = 0 given that one of the roots is x = -i \n" ); document.write( "

Algebra.Com's Answer #598627 by anand429(138) $\"\"$ $\"About$

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Given
\n" ); document.write( " $\"x%5E4+%2B+4x%5E3+%2B6x%5E2+%2B+4x+%2B+5+=+0\"$
\n" ); document.write( "Since x= -i is one of the roots,
\n" ); document.write( "i.e $\"x%5E2+=+i%5E2+=+-1\"$
\n" ); document.write( "so $\"x%5E2+%2B+1\"$ is a factor (since imaginary roots occur in conjugate pairs, hence x=i is also a factor, hence this conclusion)
\n" ); document.write( "So arranging the equation to get above factor common, we get,
\n" ); document.write( " $\"x%5E4+%2B+x%5E2+%2B+4x%5E3+%2B+4x+%2B+5x%5E2+%2B+5+=+0\"$
\n" ); document.write( "=> $\"%28x%5E2%2B4x%2B5%29%2A%28x%5E2%2B1%29=0\"$
\n" ); document.write( "So we will find the roots of $\"x%5E2%2B4x%2B5\"$ using quadratic roots formula,
\n" ); document.write( " $\"x=+%28-4%2Bsqrt%284%5E2-4%2A1%2A5%29%29%2F%282%2A1%29\"$ and $\"x=+%28-4+-+sqrt%284%5E2-4%2A1%2A5%29%29%2F%282%2A1%29\"$
\n" ); document.write( "i.e $\"x=+-2%2Bi\"$ and $\"x=+-2+-i\"$
\n" ); document.write( "So, the roots are i, -i, -2+i and -2-i \n" ); document.write( "

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