Find the value of $k$, if $x-1$ is a factor of $p\left(x\right)$ in each of the following cases: 

(i) $p\left(x\right)=x^2+x+k$ (ii) $p\left(x\right)=2x^2+kx+\sqrt{2}$ (iii) $p\left(x\right)=k{x}^2-\sqrt{2}x+1$ (iv) $p\left(x\right)=k{x}^2-3x+k$

If  $x-1$ is a factor of $p\left(x\right)$, then:

$\begin{array}{rcl}x-1& =& 0\\ x& =& 1\end{array}$

Hence, $p\left(1\right)=0$

(i) For the given polynomial, the value of $k$ can be given as:

$\begin{array}{rcl}p\left(x\right)& =& x^2+x+k\\ p\left(1\right)& =& {\left(1\right)}^2+\left(1\right)+k=0\\ 1+1+k& =& 0\\ 2+k& =& 0\\ k& =& -2\end{array}$

(ii) For the given polynomial, the value of $k$ can be given as:

$\begin{array}{rcl}p\left(x\right)& =& 2x^2+kx+\sqrt{2}\\ p\left(1\right)& =& 2{\left(1\right)}^2+k\left(1\right)+\sqrt{2}=0\\ 2+k+\sqrt{2}& =& 0\\ k& =& -\left(2+\sqrt{2}\right)\end{array}$