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? A. The function is positive on (-infinity, 0) B. the function is negative on (0, 2) C. the function is negative on (2, infinity) D. the function is positive on (o, infinity)

Answer: A,B and C are true.Step-by-step explanation:let f(x) be the given polynomial with variable x such that[tex]f(x)=ax(x-2)^4[/tex]m,where a be any odd degree negative leading coefficient of f(x),x has root as 0 with multiplicity 1 and [tex](x-2)^4[/tex] has root 2 with multiplicity 4.Lets check all the optionsA. The function is positive on (-∞, 0) let x=-1∈(-∞, 0) ⇒[tex]f(x)=a(-1)(-1-2)^4=-a(-3)^4=-81a[/tex]> 0 as a is negative.∴ function is positive on (-∞, 0) .i,e. A is true.B. The function is negative on (0, 2).Let x=1 ∈(0,2) ⇒[tex]f(x)=a(1)(1-2)^4=a(-1)^4=a[/tex]< 0 as a is negative.∴ the function is negative on (0, 2) .i,e. B is true.C. The function is negative on (2, ∞) let x=3∈(2,∞) ⇒[tex]f(x)=a(3)(3-2)^4=3a(1)^4=3a[/tex]< 0 as a is negative.∴ the function is negative on (2,∞).D.The function is positive on (0, ∞) which is not true from C.