Question

Find the value of k, if x - 1 is a factor p(x) 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

(x -1) is a factor, so we substitute x = 1 in each case and solve for k by making p(1) equal to 0.
(i) $p\left(x\right)=x^2+x+k$
$p\left(1\right)=1+1+k=0\Rightarrow k=-2$
(ii) $p\left(x\right)=2x^2+kx+\sqrt{2}$
$p\left(1\right)=2\times 1^2+k\times 1+\sqrt{2}=0$
$\Rightarrow 2+k+\sqrt{2}=0$
$\Rightarrow k=-2-\sqrt{2}=-(2+\sqrt{2})$
(iii) $p\left(x\right)=k{x}^2-\sqrt{2}x+1$
$p\left(1\right)=k-\sqrt{2}+1=0$
$\Rightarrow k=\sqrt{2}-1$
(iv) $p\left(x\right)=k{x}^2-3x+k$
$p\left(1\right)=k-3+k=0\Rightarrow 2k-3=0$
$\Rightarrow k=\frac{3}{2}$