In the function f(x)=(x+5)^4 +k , for which values of k does the function have two real solutions? no real solutions? Explain?
your on the same part as me i think is it find the zeros of the polynomial function
arent the zeros -5?
To determine the values of k for which the function has two real solutions or no real solutions, we need to analyze the discriminant of the function. The discriminant is a value that indicates the number and nature of the solutions of a quadratic or higher degree equation.
The given function, f(x) = (x + 5)^4 + k, is a quartic (degree 4) polynomial. The discriminant of a quartic polynomial can be determined by examining its cubic resolvent equation.
The cubic resolvent equation is obtained by substituting y = (x + 5)^2 into the function f(x). Thus, we have:
y^2 + k = 0
Now, we can analyze the discriminant of the cubic resolvent equation. The discriminant, denoted as Δ, satisfies the following criteria:
1. If Δ > 0, the cubic equation has one real solution and two complex conjugate solutions. This implies that the quartic function also has two real solutions.
2. If Δ = 0, the cubic equation has three real and equal solutions. Thus, the quartic function has two real solutions (one being a repeated solution).
3. If Δ < 0, the cubic equation has one real solution and two non-real complex conjugate solutions. Therefore, the quartic function has no real solutions.
Since the cubic resolvent equation is y^2 + k = 0, its discriminant is Δ = 0 - 4(1)(k) = -4k.
Now, let's analyze the cases:
1. If Δ > 0 (i.e., -4k > 0), then k < 0. In this range, the function f(x) = (x + 5)^4 + k will have two real solutions.
2. If Δ = 0 (i.e., -4k = 0), then k = 0. When k = 0, the function f(x) will have two real solutions, where one solution is repeated.
3. If Δ < 0 (i.e., -4k < 0), then k > 0. In this case, the function f(x) will have no real solutions.
Therefore, for the given function f(x) = (x + 5)^4 + k:
- If k < 0, the function has two real solutions.
- If k = 0, the function has two real solutions with one being repeated.
- If k > 0, the function has no real solutions.