Chapter 3

The function \(f(x)=-0.00002 x^{3}+0.008 x^{2}-0.3 x\) \(+6.95\) models the number of annual physician visits, \(f(x),\) by a person of age \(x\) a. Graph the function for meaningful values of \(x\) and discuss what the graph reveals in terms of the variables described by the model. b. Use the zero or root feature of your graphing utility to find the age, to the nearest year, for the group that averages 13.43 annual physician visits. c. Verify part (b) using the graph of \(f\).

The equations have real roots that are rational. Use the Rational Zero Theorem to list all possible rational roots. Then graph the polynomial function in the given viewing rectangle to determine which possible rational roots are actual roots of the equation. $$ 6 x^{3}-19 x^{2}+16 x-4=0 ;[0,2,1] \text { by }[-3,2,1] $$

Use a graphing utility to determine if the division has been performed correctly Graph the function on each side of the equation in the same viewing rectangle. If the graphs do not coincide, correct the expression on the right side by using polynomial long division. Then verify your correction using the graphing utility. $$\begin{aligned} &\frac{x^{4}+6 x^{3}+6 x^{2}-10 x-3}{x^{2}+2 x-3}=x^{2}+4 x+1, x \neq-3, x \neq 1 \end{aligned}$$

The quadratic function $$ f(x)=-0.018 x^{2}+1.93 x-25.34 $$ describes the miles per gallon, \(f(x),\) of a Ford Taurus driven at \(x\) miles per hour. Suppose that you own a Ford Taurus. Describe how you can use this function to save money.

What is meant by the end behavior of a polynomial function?

Use a graphing utility to obtain a complete graph for each polynomial function in Exercises \(58-61 .\) Then determine the number of real zeros and the number of nonreal complex zeros for each function. $$ f(x)=x^{3}-6 x-9 $$

The equations have real roots that are rational. Use the Rational Zero Theorem to list all possible rational roots. Then graph the polynomial function in the given viewing rectangle to determine which possible rational roots are actual roots of the equation. $$ \begin{aligned}&2 x^{4}+7 x^{3}-4 x^{2}-27 x-18=0 ;[-4,3,1] \text { by }[-45,45,1]\end{aligned} $$

Use a graphing utility to determine if the division has been performed correctly Graph the function on each side of the equation in the same viewing rectangle. If the graphs do not coincide, correct the expression on the right side by using polynomial long division. Then verify your correction using the graphing utility. $$\frac{2 x^{3}-3 x^{2}-3 x+4}{x-1}=2 x^{2}-x+4, x \neq 1$$

Explain how to use the Leading Coefficient Test to determine the end behavior of a polynomial function.

In a hurricane, the wind pressure varies directly as the square of the wind velocity. If wind pressure is a measure of a hurricanes destructive capacity, what happens to this destructive power when the wind speed doubles?

Use a graphing utility to obtain a complete graph for each polynomial function in Exercises \(58-61 .\) Then determine the number of real zeros and the number of nonreal complex zeros for each function. $$ f(x)=3 x^{5}-2 x^{4}+6 x^{3}-4 x^{2}-24 x+16 $$

The equations have real roots that are rational. Use the Rational Zero Theorem to list all possible rational roots. Then graph the polynomial function in the given viewing rectangle to determine which possible rational roots are actual roots of the equation. $$ 4 x^{4}+4 x^{3}+7 x^{2}-x-2=0 ;[-2,2,1] \text { by }[-5,5,1] $$

In Exercises \(59-66,\) a. Find the slant asymptote of the graph of each rational function and b. Follow the seven-step strategy and use the slant asymptote to graph each rational function. $$f(x)=\frac{x^{2}-1}{x}$$

Use a graphing utility to determine if the division has been performed correctly Graph the function on each side of the equation in the same viewing rectangle. If the graphs do not coincide, correct the expression on the right side by using polynomial long division. Then verify your correction using the graphing utility. $$\frac{3 x^{4}+4 x^{3}-32 x^{2}-5 x-20}{x+4}=3 x^{3}+8 x^{2}-5, x \neq-4$$

a. Use a graphing utility to graph \(y=2 x^{2}-82 x+720\) in a standard viewing rectangle. What do you observe? b. Find the coordinates of the vertex for the given quadratic function. c. The answer to part (b) is \((20.5,-120.5) .\) Because the leading coefficient of the given function ( 2) is positive, the vertex is a minimum point on the graph. Use this fact to help find a viewing rectangle that will give a relatively complete picture of the parabola. With an axis of symmetry at \(x=20.5,\) the setting for \(x\) should extend past this, so try \(\mathrm{Xmin}=0\) and \(\mathrm{Xmax}=30 .\) The setting for \(y\) should include (and probably go below) the \(y\) -coordinate of the graphs minimum point, so try Ymin \(=-130 .\) Experiment with Ymax until your utility shows the parabola's major features. d. In general, explain how knowing the coordinates of a parabola's vertex can help determine a reasonable viewing rectangle on a graphing utility for obtaining a complete picture of the parabola.

The illumination from a light source varies inversely as the square of the distance from the light source. If you raise a lamp from 15 inches to 30 inches over your desk, what happens to the illumination?

Why is a third-degree polynomial function with a negative leading coefficient not appropriate for modeling nonnegative real-world phenomena over a long period of time?

Use a graphing utility to obtain a complete graph for each polynomial function in Exercises \(58-61 .\) Then determine the number of real zeros and the number of nonreal complex zeros for each function. $$ f(x)=3 x^{4}+4 x^{3}-7 x^{2}-2 x-3 $$

Use Descartes's Rule of Signs to determine the possible number of positive and negative real zeros of \(f(x)=\) \(3 x^{4}+5 x^{2}+2 .\) What does this mean in terms of the graph of \(f ?\) Verify your result by using a graphing utility to graph \(f\).

a. Find the slant asymptote of the graph of each rational function and b. Follow the seven-step strategy and use the slant asymptote to graph each rational function. $$f(x)=\frac{x^{2}-4}{x}$$

Which one of the following is true? a. If a trinomial in \(x\) of degree 6 is divided by a trinomial in \(x\) of degree \(3,\) the degree of the quotient is 2. b. Synthetic division could not be used to find the quotient of \(10 x^{3}-6 x^{2}+4 x-1\) and \(x-\frac{1}{2}\). c. Any problem that can be done by synthetic division can also be done by the method for long division of polynomials. d. If a polynomial long-division problem results in a remainder that is a whole number, then the divisor is a factor of the dividend.

Find the vertex for each parabola. Then determine a reasonable viewing rectangle on your graphing utility and use it to graph the quadratic function. \(y=-0.25 x^{2}+40 x\)

The heat generated by a stove element varies directly as the square of the voltage and inversely as the resistance. If the voltage remains constant, what needs to be done to triple the amount of heat generated?

What are the zeros of a polynomial function and how are they found?

Use a graphing utility to obtain a complete graph for each polynomial function in Exercises \(58-61 .\) Then determine the number of real zeros and the number of nonreal complex zeros for each function. $$ f(x)=x^{6}-64 $$

Use Descartes's Rule of Signs to determine the possible number of positive and negative real zeros of \(f(x)=\) \(x^{5}-x^{4}+x^{3}-x^{2}+x-8 .\) Verify your result by using a graphing utility to graph \(f\).

a. Find the slant asymptote of the graph of each rational function and b. Follow the seven-step strategy and use the slant asymptote to graph each rational function. $$f(x)=\frac{x^{2}+1}{x}$$

Find \(k\) so that \(4 x+3\) is a factor of $$20 x^{3}+23 x^{2}-10 x+k$$

Find the vertex for each parabola. Then determine a reasonable viewing rectangle on your graphing utility and use it to graph the quadratic function. \(y=-4 x^{2}+20 x+160\)

Explain the relationship between the multiplicity of a zero and whether or not the graph crosses or touches the \(x\) -axis at that zero.

Galileo's telescope brought about revolutionary changes in astronomy. A comparable leap in our ability to observe the universe took place as a result of the Hubble Space Telescope. The space telescope can see stars and galaxies whose brightness is \(\frac{1}{50}\) of the faintest objects now observable using ground-based telescopes. Use the fact that the brightness of a point source, such as a star, varies inversely as the square of its distance from an observer to show that the space telescope can see about seven times farther than a ground-based telescope.

Determine a number of polynomial functions of odd degree and graph each function. Is it possible for the graph to have no real zeros? Explain. Try doing the same thing for polynomial functions of even degree. Now is it possible to have no real zeros?

a. Find the slant asymptote of the graph of each rational function and b. Follow the seven-step strategy and use the slant asymptote to graph each rational function. $$f(x)=\frac{x^{2}+4}{x}$$

When \(2 x^{2}-7 x+9\) is divided by a polynomial, the quotient is \(2 x-3\) and the remainder is \(3 .\) Find the polynomial.

Find the vertex for each parabola. Then determine a reasonable viewing rectangle on your graphing utility and use it to graph the quadratic function. \(y=5 x^{2}+40 x+600\)

Begin by deciding on a product that interests the group because you are now in charge of advertising this product. Members were told that the demand for the product varies directly as the amount spent on advertising and inversely as the price of the product. However, as more money is spent on advertising, the price of your product rises. Under what conditions would members recommend an increased expense in advertising? Once you've determined what your product is, write formulas for the given conditions and experiment with hypothetical numbers. What other factors might you take into consideration in terms of your recommendation? How do these factor affect the demand for your product?

Explain the relationship between the degree of a polynomial function and the number of turning points on its graph.

Which one of the following is true? a. The equation \(x^{3}+5 x^{2}+6 x+1=0\) has one positive real root. b. Descartes's Rule of Signs gives the exact number of positive and negative real roots for a polynomial equation. c. Every polynomial equation of degree 3 has at least one rational root. d. None of the above is true.

a. Find the slant asymptote of the graph of each rational function and b. Follow the seven-step strategy and use the slant asymptote to graph each rational function. $$f(x)=\frac{x^{2}+x-6}{x-3}$$

Find the quotient of \(x^{3 n}+1\) and \(x^{n}+1\).

Find the vertex for each parabola. Then determine a reasonable viewing rectangle on your graphing utility and use it to graph the quadratic function. \(y=0.01 x^{2}+0.6 x+100\)

Can the graph of a polynomial function have no \(x\) -intercepts? Explain.

Give an example of a polynomial equation that has no real roots. Describe how you obtained the equation.

a. Find the slant asymptote of the graph of each rational function and b. Follow the seven-step strategy and use the slant asymptote to graph each rational function. $$f(x)=\frac{x^{2}-x+1}{x-1}$$

Synthetic division is a process for dividing a polynomial by \(x-c .\) The coefficient of \(x\) is \(1 .\) How might synthetic division be used if you are dividing by \(2 x-4 ?\)

The function \(y=0.011 x^{2}-0.097 x+4.1\) models the number of people in the United States, \(y,\) in millions, holding more than one job \(x\) years after \(1970 .\) Use a graphing utility to graph the function in a \([0,20,1]\) by \([3,6,1]\) viewing rectangle. \([\text { TRACE }]\) along the curve or use your utility's minimum value feature to approximate the coordinates of the parabola's vertex. Describe what this represents in practical terms.

Can the graph of a polynomial function have no \(y\) -intercept? Explain,

a. Find the slant asymptote of the graph of each rational function and b. Follow the seven-step strategy and use the slant asymptote to graph each rational function. $$f(x)=\frac{x^{3}+1}{x^{2}+2 x}$$

Describe a strategy for graphing a polynomial function. In your description, mention intercepts, the polynomials degree, and turning points.

Explain why nonreal complex zeros are gained or lost in pairs in terms of graphs of polynomial functions.