1.

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+k x+sqrt(2)`(iii) `p(x)=k x^2-sqrt(2)x+1`(iv) `p(x)=k x^2-3x+k`

Answer» Here, `(x-1)` is a factor of ` p(x)` that means from Factor theorem,
`p(1) = 0.`
`(i) p(x) = x^2+x+k`
As, `p(1)` is `0`
`0 = 1^2+1+k=>k=-2`

`(ii) p(x) = 2x^2+kx+sqrt2`
As, `p(1)` is `0`
`2+k+sqrt2 = 0=>k = -(2+sqrt(2))`

`(iii) p(x) = kx^2-sqrt2x+1`
As, `p(1)` is `0`
`k-sqrt2+1 = 0=>k = sqrt2-1`

`(iv) p(x) = kx^2-3x+k`
As, `p(1)` is `0`
`k-3+k=0 =>2k = 3`
`k=3/2`


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