Chapter 3

Describe how to use Descartes's Rule of Signs to determine the possible number of positive real zeros of a polynomial function.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Every time I divide polynomials using synthetic division, I am using a highly condensed form of the long division procedure where omitting the variables and exponents does not involve the loss of any essential data.

Solve each inequality in Exercises \(65-70\) and graph the solution set on a real number line. $$ \frac{1}{x+1}>\frac{2}{x-1} $$

You have 80 yards of fencing to enclose a rectangular region. Find the dimensions of the rectangle that maximize the enclosed area. What is the maximum area?

Describe how to use Descartes's Rule of Signs to determine the possible number of negative roots of a polynomial equation.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. The only nongraphic method that I have for evaluating a function at a given value is to substitute that value into the function's equation.

Solve each inequality in Exercises \(65-70\) and graph the solution set on a real number line. $$ \frac{x^{2}-x-2}{x^{2}-4 x+3}>0 $$

A rectangular playground is to be fenced off and divided in two by another fence parallel to one side of the playground. Six hundred feet of fencing is used. Find the dimensions of the playground that maximize the total enclosed area. What is the maximum area?

Why must every polynomial equation with real coefficients of degree 3 have at least one real root?

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I found the zeros of function \(f,\) but 1 still need to find the solutions of the equation \(f(x)=0\)

Solve each inequality in Exercises \(65-70\) and graph the solution set on a real number line. $$ \frac{x^{2}-3 x+2}{x^{2}-2 x-3}>0 $$

A rectangular playground is to be fenced off and divided in two by another fence parallel to one side of the playground. Four hundred feet of fencing is used. Find the dimensions of the playground that maximize the total enclosed area. What is the maximum area?

Explain why the equation \(x^{4}+6 x^{2}+2-0\) has no rational roots

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. 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

Suppose \(\frac{3}{4}\) is a root of a polynomial equation. What does this tell us about the leading coefficient and the constant term in the equation?

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Synthetic division can be used to find the quotient of \(10 x^{3}-6 x^{2}+4 x-1\) and \(x-\frac{1}{2}\)

A rain gutter is made from sheets of aluminum that are 12 inches wide by turning up the edges to form right angles. Determine the depth of the gutter that will maximize its cross-sectional area and allow the greatest amount of water to flow. What is the maximum cross-sectional area?

The equations in Exercises \(72-75\) 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. $$ 2 x^{3}-15 x^{2}+22 x+15-0 ;[-1,6,1] \text { by }[-50,50,10] $$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Any problem that can be done by synthetic division can also be done by the method for long division of polynomials.

Follow the seven steps on page 399 to graph each rational function. $$f(x)=\frac{x-2}{x^{2}-4}$$

Hunky Beef, a local sandwich store, has a fixed weekly cost of \(\$ 525.00,\) and variable costs for making a roast beef sandwich are \(\$ 0.55\) a. Let \(x\) represent the number of roast beef sandwiches made and sold each week. Write the weekly cost function, C. for Hunky Beef. (Hint: The cost function is the sum of fixed and variable costs.) b. The function \(R(x)--0.001 x^{2}+3 x\) describes the money, in dollars, that Hunky Beef takes in each week from the sale of \(x\) roast beef sandwiches. Use this revenue function and the cost function from part (a) to write the store's weekly profit function, \(P\). (Hint: The profit function is the difference between the revenue and cost functions) c. Use the store's profit function to determine the number of roast beef sandwiches it should make and sell each week to maximize profit. What is the maximum weekly profit?

The equations in Exercises \(72-75\) 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. $$ 6 x^{3}-19 x^{2}+16 x-4-0 ;[0,2,1] \text { by }[-3,2,1] $$

Any problem that can be done by synthetic division can also be done by the method for long division of polynomials. If a polynomial long-division problem results in a remainder that is a whole number, then the divisor is a factor of the dividend.

What is a quadratic function?

The equations in Exercises \(72-75\) 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. $$ 2 x^{4}+7 x^{3}-4 x^{2}-27 x-18-0 ;[-4,3,1] \text { by }[-45,45,15] $$

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

What is a parabola? Describe its shape.

The equations in Exercises \(72-75\) 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. $$ 4 x^{4}+4 x^{3}+7 x^{2}-x-2-0 ;[-2,2,1] \text { by }[-5,5,1] $$

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.

Use the position function $$ s(t)--16 t^{2}+v_{0} t+s_{0} $$ \(\left(v_{11}=\text { initial velocity, } s_{0}-\text { initial position, } t-\text { time }\right)\) to answer Exercises \(75-76\) You throw a ball straight up from a rooftop 160 feet high with an initial velocity of 48 feet per second. During which time period will the ball's height exceed that of the rooftop?

Explain how to decide whether a parabola opens upward or downward.

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 \(f\)

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

Describe how to find a parabola's vertex if its equation is expressed in standard form. Give an example.

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 \(f\)

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

Describe how to find a parabola's vertex if its equation is in the form \(f(x)=a x^{2}+b x+c .\) Use \(f(x)= x^{2}-6 x+8\) as an example.

Write equations for several 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?

Use synthetic division to show that 5 is a solution of $$ x^{4}-4 x^{3}-9 x^{2}+16 x+20=0 $$ Then solve the polynomial equation.

The perimeter of a rectangle is 50 feet. Describe the possible lengths of a side if the area of the rectangle is not to exceed 114 square feet.

Use a graphing utility to obtain a complete graph for each polynomial function in Exercises \(79-82 .\) Then determine the number of real zeros and the number of imaginary zeros for each function. $$ f(x)-x^{3}-6 x-9 $$

This will help you prepare for the material covered in the next section. $$ \text { Solve: } x^{2}+4 x-1=0 $$

The perimeter of a rectangle is 180 feet. Describe the possible lengths of a side if the area of the rectangle is not to exceed 800 square feet.

Use a graphing utility to obtain a complete graph for each polynomial function in Exercises \(79-82 .\) Then determine the number of real zeros and the number of imaginary zeros for each function. $$ f(x)-3 x^{5}-2 x^{4}+6 x^{3}-4 x^{2}-24 x+16 $$

This will help you prepare for the material covered in the next section. $$ \text { Solve: } x^{2}+4 x+6=0 $$

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. $$f(x)=\frac{x^{2}-1}{x}$$

What is a polynomial incquality?

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, \(2,\) of the given function 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 graph's minimum y-value, 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.

This will help you prepare for the material covered in the next section. Let \(f(x)=a_{n}\left(x^{4}-3 x^{2}-4\right) .\) If \(f(3)=-150,\) determine the value of \(a_{n}\)

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. $$f(x)=\frac{x^{2}-4}{x}$$