91Ó°ÊÓ

For the following exercises, use the Intermediate Value Theorem to confirm that the given polynomial has at least one zero within the given interval. $$ f(x)=x^{3}-9 x, \quad \text { between } x=-4 \text { and } x=-2 $$

If a polynomial of degree \(n\) is divided by a binomial of degree 1, what is the degree of the quotient?

Use long division to divide. Specify the quotient and the remainder. $$\left(3 x^{2}-5 x+4\right) \div(3 x+1)$$

Use the given length and area of a rectangle to express the width algebraically. Length is \(3 x-4,\) area is \(6 x^{4}-8 x^{3}+9 x^{2}-9 x-4\)

Use the given volume and radius of a cylinder to express the height of the cylinder algebraically. Volume is \(\pi\left(4 x^{3}+12 x^{2}-15 x-50\right),\) radius is \(2 x+5.\)

Use your calculator to graph the polynomial function. Based on the graph, find the rational zeros. All real solutions are rational. $$f(x)=6 x^{3}-7 x^{2}+1$$

Find the dimensions of the right circular cylinder described. The radius is 3 inches more than the height. The volume is 16\(\pi\) cubic meters.

What is the fundamental difference in the algebraic representation of a polynomial function and a rational function?

Can a graph of a rational function have no \(x\) -intercepts? If so, how?

For the following exercises, find the \(x\) - and \(y\) -intercepts for the functions. $$f(x)=\frac{x}{x^{2}-x}$$