Question

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

(c) Monic of least possible degree with 3 and $\mathbf{4}\mathit{i}\mathbf{‐}\mathbf{1}$as a root.

Short answer

The monic of $\underline{the}$ least possible degree$\_$with3 and $4i-1$ as a root is $x^3-x^2-11x-51$.

Step-by-step solution

Definition of mMonic Polynomial

A monic polynomial is a single variable polynomial whose leading coefficient is equal to 1.

Finding t The monic of the least possible degree.

As given in question$\underline{,}$ one root is $4i-1$.

Taking another root as a conjugate of$4i-1$, $\sout{wehave}$the other root is role="math" localid="1653644520929" $‐4i‐1$.

Therefore, $\_$the required monic is given as:

$$\begin{gathered} Themonic=\left(x-3\right)\left(x‐\left(4i‐1\right)\right)\left(x‐\left(‐4i‐1\right)\right) \\ =\left(x-3\right)\left(x‐4i+1\right)\left(x+4i+1\right) \\ =\left(x-3\right)\left(x^2+2x+1+16\right) \\ =\left(x-3\right)\left(x^2+2x+17\right) \\ =\left(x^3‐3x^2+2x^2‐6x‐17x‐51\right) \\ =x^3‐x^2‐11x‐51 \end{gathered}$$

Therefore, the $\underline{\mathbf{m}}\sout{\mathbf{M}}$onic of least possible degree with 3 and $4i‐1$as a root is: $x^3‐x^2‐11x‐51$.