a polynomial function has a root of 0 with multiplicity 1, and a root of 2 with multiplicity 4. if the…

a polynomial function has a root of 0 with multiplicity 1, and a root of 2 with multiplicity 4. if the function has a negative leading coefficient, and is of odd degree, which of the following are true?\nthe function is positive on (-∞, 0).\nthe function is negative on (0, 2).\nthe function is negative on (2, ∞).\nthe function is positive on (0, ∞).\ndone

Answer

Explanation:

Step1: Determine the degree of the polynomial

The root 0 has multiplicity 1 and the root 2 has multiplicity 4. So the degree of the polynomial $n=1 + 4=5$ (an odd - degree polynomial). Let the polynomial be $f(x)=a(x - 0)^1(x - 2)^4$, where $a<0$ (negative leading coefficient).

Step2: Analyze the sign of the function on different intervals

For $x<0$, let's take a test - point, say $x=-1$. Then $f(-1)=a(-1)(-1 - 2)^4=a(-1)(81)=-81a$. Since $a < 0$, $-81a>0$. So the function is positive on $(-\infty,0)$.
For $0<x<2$, let's take a test - point, say $x = 1$. Then $f(1)=a(1)(1 - 2)^4=a(1)(1)=a<0$. So the function is negative on $(0,2)$.
For $x>2$, let's take a test - point, say $x = 3$. Then $f(3)=a(3)(3 - 2)^4=3a<0$. So the function is negative on $(2,\infty)$.

Answer:

The function is positive on $(-\infty,0)$. The function is negative on $(0,2)$. The function is negative on $(2,\infty)$.