Question

How do you find k such that `f(x) = x^4 - kx^3 + kx^2 + 1` has the factor x + 2?

Answer 1

There is no Real value for `k` such that `x+2` is a factor of `f(x)=x^4-kx^3+kx^2+1`

Explanation:

If `(x+2)` is a factor of `f(x)=x^4-kx^3+kx^2+1`
then
`color(white)("XXX")f(x)=(x+2)(g(x))` for some polynomial `g(x)`
from which it is clear that if `x=(-2)` then `f(x)=0`

Substituting `(-2)` for `x` and noting that the result is equal to zero:
`color(white)("XXX")(-2)^2-(-2k)^3+(-2k)^2+1=0`

`color(white)("XXX")16k^4+8k^3+4k^2+1=0`

At (local) minimum/maximum points the slope of
`color(white)("XXX")h(k)=18k^4+8k^3+4k^2+1` must be equal to zero.
Looking for values of `k` such that
`color(white)("XXX")(d(h(k)))/(dk) = 64k^3+24k^2+8k=0`

`color(white)("XXX")8(k)(8k^2+3k+8)=0`

`color(white)("XXX")rArr k=0 or (8k^2+3k+1)=0`
`color(white)("XXX")`...but applying the quadratic formula we can see that
`color(white)("XXXXXX")(k^2+3k+1)=0` has no Real solutions.
So the only critical point occurs when `k=0`
and `f(x)` has a minimum value when `k=0`

Since `f(x) > 0` for all Real values of `x` when `k=0`
`color(white)("XXX")`(since, with `k=0`, `f(x)=x^4+1`)

Therefore there is no value of `k` for which `(x+2)` is a factor of the given expression.