Question

Find a polynomial in $\mathbf{\square }\left[x\right]$that satisfies the given conditions$\left(b\right)Monicofleastpossibledegreewith$$\mathbf{1}\mathbf{-}\mathit{i}$and $\mathbf{2}\mathit{i}$$asroots$.

Short answer

The required monic polynomials of the least possible degree is:$x^4-2x^3+6x^2-8x+8$

Step-by-step solution

Use the Lemma 4.29

According to the statement, if $f\left(x\right)$is a polynomial in $\square \left[x\right]$and $x+iy$is a root of $f\left(x\right)$in set of complex numbers$-$, then $x-iy$is also a root of $f\left(x\right)$.

By using this lemma, we can say that $1+i$and$-2i$ is also a root of$f\left(x\right)$

Step 2:_Find the required polynomial

Now, the least possible degree of a monic polynomial with these roots is 4.

Also, here the roots of $f\left(x\right)$are$1-i,1+i,2i,-2i$

$$\begin{gathered} \Rightarrow f\left(x\right)=\left(x-2i\right)\left(x+2i\right)\left(x-\left(1+i\right)\right)\left(x-\left(1-i\right)\right) \\ \Rightarrow f\left(x\right)=\left(x^2+4\right)\left(x^2-2x+2\right) \\ \Rightarrow f\left(x\right)=x^4-2x^3+6x^2-8x+8 \end{gathered}$$

Therefore, our desired polynomial is $x^4-2x^3+6x^2-8x+8$.