Find the value of k, if x-1 is a factor of p(x) in each of the following cases :
(i) `p(x)=x^(2)+x+k " " (ii) p(x)=2x^(2)+kx+sqrt(2)`
(iii) `p(x)=kx^(2)-sqrt(2)x+1 " " (iv) p(x) =kx^(2)-3x+k`

Explanation

The zero of x-1 is 1.
(i) `because (x-1)` is a factor of p(x), then p(1)=0
`implies 1^(2)+1+k=0 " " [because p(x)=x^(2)+x+k]`
`implies 2+k=0`
`implies k=-2`
(ii) `because (x-1)` is a factor of p(x), then p(1)=0
`implies 2(1)^(2)+k(1)+sqrt(2)=0 " " [because p(x)=2x^(2)+kx+sqrt(2)]`
`implies 2+k+sqrt(2)=0`
`implies k=-(2+sqrt(2))`
(iii) `because (x-1)` is a factor of p(x), then p(1)=0
`implies k(1)^(2)-sqrt(2)+1=0 " " [because p(x)=kx^(2)-sqrt(2)+1]`
`implies k=(sqrt(2)-1)`
(iv) `because (x-1)` is a factor of p(x), then p(1)=0
`implies k(1)^(2)-3+k=0 " " [because p(x)=kx^(2)-3x+k]`
`implies 2k-3=0`
`implies k=(3)/(2)`