91Ó°ÊÓ

In Exercises 31 - 36, solve the inequality and write the solution set in interval notation. \( x^3 - 4x \ge 0 \)

In Exercises 31- 34, use a graphing utility to graph the functions \( f \) and \( g \) in the same viewing window. Zoom out sufficiently far to show that the right-hand and left-hand behaviors of \( f \) and \( g \) appear identical. \( f(x) = -(x^4 - 4x^3 + 16x) \), \( g(x) = -x^4 \)

In Exercises 35- 50, (a) find all the real zeros of the polynomial function, (b) determine the multiplicity of each zero and the number of turning points of the graph of the function, and (c) use a graphing utility to graph the function and verify your answers. \( h(t) = t^2 - 6t + 9 \)

In Exercises 47 - 54, write the function in the form \( f(x) = (x - k)q(x) + r \) for the given value of \( k \), demonstrate that \( f(k) = r \). \( f(x) = x^3 - x^2 - 14x + 11 \) , \( k = 4 \)

In Exercises 35- 50, (a) find all the real zeros of the polynomial function, (b) determine the multiplicity of each zero and the number of turning points of the graph of the function, and (c) use a graphing utility to graph the function and verify your answers. \( f(x) = 2x^4 - 2x^2 - 40 \)

In Exercises 51 - 54, write the polynomial (a) as the product of factors that are irreducible over the rationals, (b) as the product of linear and quadratic factors that are irreducible over the reals, and (c) in completely factored form. \( f(x) = x^4 - 3x^3 - x^2 - 12x - 20 \) (Hint: One factor is \( x^2 + 4 \).)

In Exercises 55 - 58, use a graphing utility to graph the equation. Use the graph to approximate the values of \( x \) that satisfy each inequality. Equation \( y = \dfrac{3x}{x - 2} \) Inequalities \( (a) \) y \le 0 \( (b) \) y \ge 6 $

In Exercises 55 - 58, use the Remainder Theorem and synthetic division to find each function value. Verify your answers using another method. \( f(x) = 2x^3 - 7x + 3 \) (a) \( f(1) \) (b) \( f(-2) \) (c) \( f\left(\frac{1}{2}\right) \) (d) \( f(2) \)

In Exercises 59 - 66, use synthetic division to show that \( x \) is a solution of the third-degree polynomial equation, and use the result to factor the polynomial completely. List all real solutions of the equation. \( x^3 - 28x - 48 = 0 \), \( x = -4 \)

In Exercises 59 - 66, use synthetic division to show that \( x \) is a solution of the third-degree polynomial equation, and use the result to factor the polynomial completely. List all real solutions of the equation. \( 2x^3 - 15x^2 + 27x - 10 = 0 \), \( x = \frac{1}{2} \)