Question

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) $p\left(x\right)=k{x}^2-3x+k$

Answer 1

If $x-1$ is the factor of the equation then put $x=1$ in these equation

(i) $x^2+x+k=0\Rightarrow (1{)}^2+1+k=0\Rightarrow 1+1+k=0\Rightarrow k=-2$

(ii) $2x^2+kx+\sqrt{2}=0\Rightarrow 2(1{)}^2+k(1)+\sqrt{2}=0\Rightarrow 2+k+\sqrt{2}=0\Rightarrow k=-2-\sqrt{2}$

(iii) $k{x}^2-\sqrt{2}x+1=0\Rightarrow k(1{)}^2-\sqrt{2}(1)+1=0\Rightarrow k-\sqrt{2}+1=0\Rightarrow k=\sqrt{2}-1$

(iv) $k{x}^2-3x+k=0\Rightarrow k(1{)}^2-3(1)+k=0\Rightarrow k-3+k=0\Rightarrow 2k=-3\Rightarrow k=-\frac{3}{2}$