Question

if $h\left(x\right)$is a factor of $f\left(x\right)$.

1. For what value of $k$is $x-2$a factor of $x^4-5x^3+5x^2+3x+k\mathrm{in}\mathrm{\mathbb{Q}}\left[x\right]$?
2. For what value of $k$is $x+1$a factor of ${\mathrm{x}}^4+2{\mathrm{x}}^3-3{\mathrm{x}}^2+\mathrm{kx}+1\mathrm{in}{\mathrm{\mathbb{Z}}}_5\left[\mathrm{x}\right]$?

Short answer

1. It is proved that the required value is$k=-2$ .
2. It is proved that the required value is $k=2$.

Step-by-step solution

Factor Theorem:

If $F$ is any field such that$f\left(x\right)\in F\left[x\right]$and $k\in F$. Then, $k$will be the root of the polynomial$f\left(x\right)$ , if and only if $\left(x-k\right)$ is a factor of $f\left(x\right)$in $F\left[x\right]$.

Finding missing term:

(a)

Given, $f_k\left(x\right)=x^4-5x^3+5x^2+3x+k$

At $x=2$, we have:$f_k\left(2\right)=k+2$

Since, $x-2$ is a factor of $f\left(x\right)$. This implies that $x=2$, we have:

$\begin{array}{c}{f}_k\left(2\right)=0\\ k+2=0\\ k=-2\end{array}$

Hence, the required value is $k=-2$.

Finding missing term:

(b)

Given, $f_k\left(x\right)=x^4+2x^3-3x^2+kx+1$

At $x=-1$, we have:$f_k\left(-1\right)=-k-3$

Since, $x+1$ is a factor of $f\left(x\right)$. This implies that $x=-1$, we have:

$\begin{array}{c}{f}_k\left(-1\right)=0\\ -k-3=0\\ k=-3=2\end{array}$

Hence, the required value is $k=2$.