f(x)=x64-4x^2+k. Find the values of k so thath the zeros of f satisfy the specified charachteristics.

Question

a.four real zeros
b.two real zeros, each of multiplicity of two
c.two real zeros and two complex zeros
d.four complez zeros

a.I don't know if this range goes for this or b. 1<x<3?
b. k=4 and x=+ √2 and x=- √2
And I don't know how to get c and d
any help would be appreciated :)

Answer 1

I gotta think you meant

f(x) = x^4-4x^2+k

we know that
x^4-4x^2 = x^2(x^2-4) = x^2(x-2)(x+2)
so it has 3 real roots, one repeated.

x^4-4x^2+4 = (x^2-4)^2 = (x-2)^2(x+2)^2
so it has 4 real roots (2 distinct)

x^4-4x^2+k has 4 distinct real roots if 0 < k < 4

x^4-4x^2+k has 4 complex roots if k>4

x^4-4x^2+k has two real, two complex roots if k < 0

To convince yourself of this, just check the graphs of the first two functions.