Answer: The correct option is D, i.e., The graph of function is negative on \((3,\infty)\).
Explanation:
If a polynomial have a root p with multiplicity q, then \((x-p)^q\) is a factor of the polynomial.
It is given polynomial function has a root of –5 with multiplicity 3, a root of 1 with multiplicity 2, and a root of 3 with multiplicity 7. The leading coefficient is negative.
From the given information the polynomial is,
\(P(x)=-(x+5)^3(x-1)^2(x-3)^7\)
The graph of the polynomial shown the figure.
Since the leading coefficient is negative, therefore \(p(x)\rightarrow -\infty\) as \(x\rightarrow -\infty\) and \(p(x)\rightarrow -\infty\) as \(x\rightarrow \infty\).
The 3 roots divides the number line in 4 intervals \((-\infty,-5),(-5,1),(1,3),(3,\infty)\).
From figure it is noticed that the graph of the function is negative on \((3,\infty)\).
