Chapter 3

For the following exercises, identify the function as a power function, a polynomial function, or neither. $$f(x)=\frac{x^{2}}{x^{2}-1}$$

For the following exercises, identify the function as a power function, a polynomial function, or neither. $$f(x)=2 x(x+2)(x-1)^{2}$$

For the following exercises, identify the function as a power function, a polynomial function, or neither. $$f(x)=3^{x+1}$$

For the following exercises, find the degree and leading coefficient for the given polynomial. $$-3 x$$

For the following exercises, find the degree and leading coefficient for the given polynomial. $$7-2 x^{2}$$

For the following exercises, find the degree and leading coefficient for the given polynomial. $$-2 x^{2}-3 x^{5}+x-6$$

For the following exercises, find the degree and leading coefficient for the given polynomial. $$x\left(4-x^{2}\right)(2 x+1)$$

For the following exercises, find the degree and leading coefficient for the given polynomial. $$x^{2}(2 x-3)^{2}$$

For the following exercises, determine the end behavior of the functions. $$f(x)=x^{4}$$

For the following exercises, determine the end behavior of the functions. $$f(x)=x^{3}$$

For the following exercises, determine the end behavior of the functions. $$f(x)=-x^{4}$$

For the following exercises, determine the end behavior of the functions. $$f(x)=-x^{9}$$

For the following exercises, determine the end behavior of the functions. $$f(x)=-2 x^{4}-3 x^{2}+x-1$$

For the following exercises, determine the end behavior of the functions. $$f(x)=3 x^{2}+x-2$$

For the following exercises, determine the end behavior of the functions. $$f(x)=x^{2}\left(2 x^{3}-x+1\right)$$

For the following exercises, determine the end behavior of the functions. $$f(x)=(2-x)^{7}$$

For the following exercises, find the intercepts of the functions. $$f(t)=2(t-1)(t+2)(t-3)$$

For the following exercises, find the intercepts of the functions. $$g(n)=-2(3 n-1)(2 n+1)$$

For the following exercises, find the intercepts of the functions. $$f(x)=x^{4}-16$$

For the following exercises, find the intercepts of the functions. $$f(x)=x^{3}+27$$

For the following exercises, find the intercepts of the functions. $$f(x)=x\left(x^{2}-2 x-8\right)$$

For the following exercises, find the intercepts of the functions. $$f(x)=(x+3)\left(4 x^{2}-1\right)$$

For the following exercises, make a table to confirm the end behavior of the function. $$f(x)=-x^{3}$$

For the following exercises, make a table to confirm the end behavior of the function. $$f(x)=x^{4}-5 x^{2}$$

For the following exercises, make a table to confirm the end behavior of the function. $$f(x)=x^{2}(1-x)^{2}$$

For the following exercises, make a table to confirm the end behavior of the function. $$f(x)=\frac{x^{5}}{10}-x^{4}$$

For the following exercises, graph the polynomial functions using a calculator. Based on the graph, determine the intercepts and the end behavior. $$f(x)=x^{3}(x-2)$$

For the following exercises, graph the polynomial functions using a calculator. Based on the graph, determine the intercepts and the end behavior. $$f(x)=x(14-2 x)(10-2 x)$$

For the following exercises, graph the polynomial functions using a calculator. Based on the graph, determine the intercepts and the end behavior. $$f(x)=x(14-2 x)(10-2 x)^{2}$$

For the following exercises, graph the polynomial functions using a calculator. Based on the graph, determine the intercepts and the end behavior. $$f(x)=x^{3}-16 x$$

For the following exercises, graph the polynomial functions using a calculator. Based on the graph, determine the intercepts and the end behavior. $$f(x)=x^{3}-27$$

For the following exercises, graph the polynomial functions using a calculator. Based on the graph, determine the intercepts and the end behavior. $$f(x)=x^{4}-81$$

For the following exercises, graph the polynomial functions using a calculator. Based on the graph, determine the intercepts and the end behavior. $$f(x)=-x^{3}+x^{2}+2 x$$

For the following exercises, graph the polynomial functions using a calculator. Based on the graph, determine the intercepts and the end behavior. $$f(x)=x^{3}-2 x^{2}-15 x$$

For the following exercises, graph the polynomial functions using a calculator. Based on the graph, determine the intercepts and the end behavior. $$f(x)=x^{3}-0.01 x$$

For the following exercises, use the information about the graph of a polynomial function to determine the function. Assume the leading coefficient is 1 or –1. There may be more than one correct answer. The \(y\) - intercept is \((0,-4) .\) The \(x\) - intercepts are \((-2,0),(2,0) .\) Degree is \(2 .\) End behavior: as \(x \rightarrow-\infty, f(x) \rightarrow \infty,\) as \(x \rightarrow \infty, f(x) \rightarrow \infty\)

For the following exercises, use the information about the graph of a polynomial function to determine the function. Assume the leading coefficient is 1 or –1. There may be more than one correct answer. The \(y\) - intercept is \((0,9) .\) The \(x\) - intercepts are \((-3,0),(3,0) .\) Degree is \(2 .\) End behavior: as \(x \rightarrow-\infty, f(x) \rightarrow-\infty,\) as \(x \rightarrow \infty, f(x) \rightarrow-\infty\)

For the following exercises, use the information about the graph of a polynomial function to determine the function. Assume the leading coefficient is 1 or –1. There may be more than one correct answer. The \(y\) - intercept is \((0,0) .\) The \(x\) - intercepts are \((0,0),(2,0) .\) Degree is \(3 .\) End behavior: as \(x \rightarrow-\infty, f(x) \rightarrow-\infty,\) as \(x \rightarrow \infty, f(x) \rightarrow \infty\)

For the following exercises, use the information about the graph of a polynomial function to determine the function. Assume the leading coefficient is 1 or –1. There may be more than one correct answer. The \(y\) - intercept is \((0,1) .\) The \(x\) -intercept is \((1,0) .\) Degree is \(3 .\) End behavior: as \(x \rightarrow-\infty, f(x) \rightarrow \infty,\) as \(x \rightarrow \infty, f(x) \rightarrow-\infty\).

For the following exercises, use the written statements to construct a polynomial function that represents the required information. An oil slick is expanding as a circle. The radius of the circle is increasing at the rate of 20 meters per day. Express the area of the circle as a function of \(d,\) the number of days elapsed.

For the following exercises, use the written statements to construct a polynomial function that represents the required information. A cube has an edge of 3 feet. The edge is increasing at the rate of 2 feet per minute. Express the volume of the cube as a function of \(m\), the number of minutes elapsed.

For the following exercises, use the written statements to construct a polynomial function that represents the required information. A rectangle has a length of 10 inches and a width of 6 inches. If the length is increased by \(x\) inches and the width increased by twice that amount, express the rectangle as a function of \(x .\)

For the following exercises, use the written statements to construct a polynomial function that represents the required information. An open box is to be constructed by cutting out square comers of \(x\) - inch sides from a piece of cardboard 8 inches by 8 inches and then folding up the sides. Express the volume of the box as a function of \(x .\)

For the following exercises, use the written statements to construct a polynomial function that represents the required information. A rectangle is twice as long as it is wide. Squares of side 2 feet are cut out from each corner. Then the sides are folded up to make an open box. Express the volume of the box as a function of the width \((x)\).

What is the difference between an \(x\) - intercept and a zero of a polynomial function \(f ?\)

If a polynomial function of degree \(n\) has \(n\) distinct zeros, what do you know about the graph of the function?

Explain how the Intermediate Value Theorem can assist us in finding a zero of a function.

Explain how the factored form of the polynomial helps us in graphing it.

If the graph of a polynomial just touches the \(x\) -axis and then changes direction, what can we conclude about the factored form of the polynomial?

For the following exercises, find the \(x\) - or \(t\) -intercepts of the polynomial functions. $$ C(t)=2(t-4)(t+1)(t-6) $$