Question

What are the values of `k` when `x^4-3x^2+x-2` is divided by `x-k` and has a remainder of `2`?

Answer 1

`k=-2.094945` and `1.893974`

Explanation:

According to remainder theorem , when we divide a polynomial `f(x)` by `x-k` remainder is `f(k)`.

Therefore dividing `x^4-3x^2+x-2` by `x-k` we get

`k^4-3k^2+k-2` and as remainder is `2`, we have

`k^4-3k^2+k-2=2` or `k^4-3k^2+k-4=0`

Further according to rational zeros theorem , If `f(x)` is a polynomial with integer coefficients and if `p/q` is a zero of `f(x)` i.e. `f(p/q)=0`, then `p` is a factor of the constant term of `f(x)` and `q` is a factor of the leading coefficient of `f(x)`.

The factors of `-4` are `+-1,+-2,+-4` and as for none of them as `k`, `k^4-3k^2+k-4=0`, we do not have rational roots.

The graph appears like this and solutions are slightly less than `-2` and `2`. These values obtained using Goal Seek in MS Excel are `-2.094945` and `1.893974`.

graph{x^4-3x^2+x-4 [-3, 3, -5, 5]}