Question

On the basis of the graph, answers the following questions.

Answer 1

a) The factors have the form:

$(x-x_1)$

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.