Question

The change in water level of a lake is modeled by a polynomial function, W(x). Describe how to find the x-intercepts of W(x) and how to construct a rough graph of W(x) so that the Parks Department can predict when there will be no change in the water level. You may create a sample polynomial of degree 3 or higher to use in your explanations.

(Dividing and Solving Polynomials)

Answer 1

First. Finding the x-intercepts of

Let be the change in water level. So to find the x-intercepts of this function we can use The Rational Zero Test that states:

To find the zeros of the polynomial:

We use the Trial-and-Error Method which states that a factor of the constant term:

can be a zero of a polynomial (the x-intercepts).

So let's use an example: Suppose you have the following polynomial:

where the constant term is . The possible zeros are the factors of this term, that is:

.

Thus:

From the foregoing, we can affirm that are zeros of the polynomial.

Second. Construction a rough graph of

Given that this is a polynomial, then the function is continuous. To graph it we set the roots on the coordinate system. We take the interval:

and compute where is a real number between -2 and -1. If , the curve start rising, if not, the curve start falling. For instance:

Therefore the curve start falling and it goes up and down until and from this point it rises without a bound as shown in the figure below