Answer:
C) p(x) has a zero between x=2 and x=4
Step-by-step explanation:
For a polynomial p(x), means that, when p(x) is divided by
, the remainder is
.
This is called the remainder theorem
Therefore ,
means that when p(x) is divided
and
is 1 and
respectively.
Also the Intermediate value theorem says that, if p(a) and p(b) have opposite signs then there is a certain on
where
i.e there is a zero between
and
Therefore the correct answer is C
a) The factors have the form:
where x1 is a zero of the function. A zero is a point at which the graph intercepts the x-axis. From the graph, the zeros are:
-6, -4, 2, and 3
Therefore, the factors are:
(x + 6)
(x + 4)
(x - 2)
(x - 3)
b) Multiplying all these factors we get a polynomial, p(x), with the zeros of the graph. That is:
p(x) = (x + 6)(x + 4)(x - 2)(x - 3)
c) Yes, it is possible to find other polynomials with the same zeros. To do that we have to multiply p(x) by a constant. For example, multiplying by 2:
f(x) = 2(x + 6)(x + 4)(x - 2)(x - 3)
and f(x) has the same zeros as p(x)
d) Every polynomial obtained in the previous way, multiplying p(x) by a constant, will have a different graph. In conclusion, it is not possible to find other polynomials with the same zeros and the same graph.
