Question

Write a polynomial function P(x) with rational coefficients so that P(x)=0 has the given roots? 41) √3, 2, -i 44) -√7, i

Answer 1

41.irrational and complex roots always occur in pairs.
\[conjugate~ of \sqrt{3}~is~-\sqrt{3}.conjugate ~of~-\iota~is~\iota \]
\[x+\sqrt{3}~and~x-\sqrt{3}~are~factors.\]
\[\left( x+\sqrt{3} \right)\left( x-\sqrt{3} \right)=x^2-3 ~is~factor~of~P(x)=0\]
similarly \[\left( x-\iota \right)\left( x+\iota\right)=x^2-\iota^2=x^2+1~is~also~factor~of~P(x)=0\]
Hence\[ P(x)=\left( x^2-3 \right)\left( x^2+1 \right)\left( x-2 \right)=0\]
multiply and get the required polynomial.

Answer 2

Thank you so much.