A 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. If the function has a negative leading coefficient and is of even degree, which statement about the graph is true?
The graph of the function is positive on (mc024-1.jpg, –5).
The graph of the function is negative on (–5, 3).
The graph of the function is positive on (mc024-2.jpg, 1).
The graph of the function is negative on (3, mc024-3.jpg).
Answers
Step-by-step explanation:
If a polynomial have a root p with multiplicity q, then (x-p)^q(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)^7P(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 -\inftyp(x)→−∞ as x\rightarrow -\inftyx→−∞ and p(x)\rightarrow -\inftyp(x)→−∞ as x\rightarrow \inftyx→∞ .
The 3 roots divides the number line in 4 intervals (-\infty,-5),(-5,1),(1,3),(3,\infty)(−∞,−5),(−5,1),(1,3),(3,∞) .
From figure it is noticed that the graph of the function is negative on (3,\infty)(3,
Answer:
The graph of the function is negative on (3, mc024-3.jpg).
Step-by-step explanation: