Question

Find a polynomial in $\mathbf{\square }\left[x\right]$that satisfies the given conditions

$\left(a\right)$Monic of degree 3 with 2 and $\mathbf{3}\mathbf{+}\mathit{i}$ as a root.

Short answer

The required monic polynomial of degree 3 is $x^3-8x^2+22x-20$.

Step-by-step solution

Use the Lemma 4.29

According to the lemma, 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 $3-i$is also a root of$f\left(x\right)$

Find the required polynomial

Let our desire monic polynomial of degree 3 be:

$f\left(x\right)=x^3+b{x}^2+cx+d$

Also, here the roots of $f\left(x\right)$are$2,3+i,3-i$

$$\begin{gathered} \Rightarrow f\left(x\right)=\left(x-2\right)\left(x-\left(3+i\right)\right)\left(x-\left(3-i\right)\right) \\ \Rightarrow f\left(x\right)=\left(x-2\right)\left(x^2-6x+10\right) \\ \Rightarrow f\left(x\right)=x^3-8x^2+22x-20 \end{gathered}$$

Therefore, our desired polynomial is $x^3-8x^2+22x-20$.