Chapter 3

In Exercises \(27-34,\) find the zeros for each polynomial function and give the multiplicity for each zero. State whether the graph crosses the \(x\) -axis, or touches the \(x\) -axis and turns around, at each zero. $$f(x)=3(x+5)(x+2)^{2}$$

Use the four-step procedure for solving variation problems given on page 356 to solve. \(a\) is directly proportional to \(b\) and inversely proportional to the square of \(c . a=7\) when \(b=9\) and \(c=6 .\) Find \(a\) when \(b=4\) and \(c=8\)

In Exercises \(23-28\), factor each polynomial: a. as the product of factors that are irreducible over the rational numbers. b. as the product of factors that are irreducible over the real numbers. c. in completely factored form involving complex nonreal, or imaginary, numbers. \(x^{4}-4 x^{3}+14 x^{2}-36 x+45\) (Hint: One factor is \(\left.x^{2}+9 .\right)\)

Use Descartes's Rule of Signs to determine the possible number of positive and negative real zeros for each given function. $$ f(x)=4 x^{4}-x^{3}+5 x^{2}-2 x-6 $$

Find the vertical asymptotes, if any, of the graph of each rational function. $$r(x)=\frac{x}{x^{2}+3}$$

Divide using synthetic division. $$\frac{x^{7}+x^{5}-10 x^{3}+12}{x+2}$$

Use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range. \(f(x)=x^{2}-2 x-15\)

In Exercises \(27-34,\) find the zeros for each polynomial function and give the multiplicity for each zero. State whether the graph crosses the \(x\) -axis, or touches the \(x\) -axis and turns around, at each zero. $$f(x)=4(x-3)(x+6)^{3}$$

In Exercises \(29-36,\) find an nth-degree polynomial function with real coefficients satisfying the given conditions. If you are using a graphing utility, use it to graph the function and verify the real zeros and the given function value. \(n=3 ; 1\) and \(5 i\) are zeros; \(f(-1)=-104\)

Find all zeros of the polynomial function or solve the given polynomial equation. Use the Rational Zero Theorem, Descartes's Rule of Signs, and possibly the graph of the polynomial function shown by a graphing utility as an aid in obtaining the first zero or the first root. $$ f(x)=x^{3}-4 x^{2}-7 x+10 $$

In Exercises \(29-36,\) find the horizontal asymptote, if any, of the graph of each rational function. $$f(x)=\frac{12 x}{3 x^{2}+1}$$

Divide using synthetic division. $$\frac{x^{4}-256}{x-4}$$

Use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range. \(f(x)=x^{2}+3 x-10\)

In Exercises \(27-34,\) find the zeros for each polynomial function and give the multiplicity for each zero. State whether the graph crosses the \(x\) -axis, or touches the \(x\) -axis and turns around, at each zero. $$f(x)=-3\left(x+\frac{1}{2}\right)(x-4)^{3}$$

In Exercises \(29-36,\) find an nth-degree polynomial function with real coefficients satisfying the given conditions. If you are using a graphing utility, use it to graph the function and verify the real zeros and the given function value. \(n=3 ; 4\) and \(2 i\) are zeros; \(f(-1)=-50\)

Find all zeros of the polynomial function or solve the given polynomial equation. Use the Rational Zero Theorem, Descartes's Rule of Signs, and possibly the graph of the polynomial function shown by a graphing utility as an aid in obtaining the first zero or the first root. $$ f(x)=x^{3}+12 x^{2}+21 x+10 $$

Find the horizontal asymptote, if any, of the graph of each rational function. $$f(x)=\frac{15 x}{3 x^{2}+1}$$

Divide using synthetic division. $$\frac{x^{7}-128}{x-2}$$

Use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range. \(f(x)=2 x^{2}-7 x-4\)

In Exercises \(27-34,\) find the zeros for each polynomial function and give the multiplicity for each zero. State whether the graph crosses the \(x\) -axis, or touches the \(x\) -axis and turns around, at each zero. $$f(x)=x^{3}-2 x^{2}+x$$

A person's fingernail growth, \(G\), in inches, varies directly as the number of weeks it has been growing, \(W\). a. Write an equation that expresses this relationship. b. Fingernails grow at a rate of about 0.02 inch per week. Substitute 0.02 for \(k,\) the constant of variation, in the equation in part (a) and write the equation for fingernail growth. c. Substitute 52 for \(W\) to determine your fingernail length at the end of one year if for some bizarre reason you decided not to cut them and they did not break.

In Exercises \(29-36,\) find an nth-degree polynomial function with real coefficients satisfying the given conditions. If you are using a graphing utility, use it to graph the function and verify the real zeros and the given function value. \(n=3 ;-5\) and \(4+3 i\) are zeros; \(f(2)=91\)

Find all zeros of the polynomial function or solve the given polynomial equation. Use the Rational Zero Theorem, Descartes's Rule of Signs, and possibly the graph of the polynomial function shown by a graphing utility as an aid in obtaining the first zero or the first root. $$ 2 x^{3}-x^{2}-9 x-4=0 $$

Find the horizontal asymptote, if any, of the graph of each rational function. $$g(x)=\frac{12 x^{2}}{3 x^{2}+1}$$

Divide using synthetic division. $$\frac{2 x^{5}-3 x^{4}+x^{3}-x^{2}+2 x-1}{x+2}$$

Use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range. \(f(x)=2 x-x^{2}+3\)

In Exercises \(27-34,\) find the zeros for each polynomial function and give the multiplicity for each zero. State whether the graph crosses the \(x\) -axis, or touches the \(x\) -axis and turns around, at each zero. $$f(x)=x^{3}+4 x^{2}+4 x$$

A person's wages, \(W,\) vary directly as the number of hours worked, \(h\) a. Write an equation that expresses this relationship. b. For a 40 -hour work week, Gloria earned \(\$ 1400\). Substitute 1400 for \(W\) and 40 for \(h\) in the equation from part (a) and find \(k,\) the constant of variation. c. Substitute the value of \(k\) into your equation in part (a) and write the equation that describes Gloria's wages in terms of the number of hours she works. d. Use the equation from part (c) to find Gloria's wages for 25 hours of work.

In Exercises \(29-36,\) find an nth-degree polynomial function with real coefficients satisfying the given conditions. If you are using a graphing utility, use it to graph the function and verify the real zeros and the given function value. \(n=3 ; 6\) and \(-5+2 i\) are zeros; \(f(2)=-636\)

Find all zeros of the polynomial function or solve the given polynomial equation. Use the Rational Zero Theorem, Descartes's Rule of Signs, and possibly the graph of the polynomial function shown by a graphing utility as an aid in obtaining the first zero or the first root. $$ 3 x^{3}-8 x^{2}-8 x+8=0 $$

Find the horizontal asymptote, if any, of the graph of each rational function. $$g(x)=\frac{15 x^{2}}{3 x^{2}+1}$$

Divide using synthetic division. $$\frac{x^{5}-2 x^{4}-x^{3}+3 x^{2}-x+1}{x-2}$$

Use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range. \(f(x)=5-4 x-x^{2}\)

In Exercises \(27-34,\) find the zeros for each polynomial function and give the multiplicity for each zero. State whether the graph crosses the \(x\) -axis, or touches the \(x\) -axis and turns around, at each zero. $$f(x)=x^{3}+7 x^{2}-4 x-28$$

In Exercises \(29-36,\) find an nth-degree polynomial function with real coefficients satisfying the given conditions. If you are using a graphing utility, use it to graph the function and verify the real zeros and the given function value. \(n=4 ; i\) and \(3 i\) are zeros; \(f(-1)=20\)

Find all zeros of the polynomial function or solve the given polynomial equation. Use the Rational Zero Theorem, Descartes's Rule of Signs, and possibly the graph of the polynomial function shown by a graphing utility as an aid in obtaining the first zero or the first root. $$ f(x)=x^{4}-2 x^{3}+x^{2}+12 x+8 $$

Find the horizontal asymptote, if any, of the graph of each rational function. $$h(x)=\frac{12 x^{3}}{3 x^{2}+1}$$

Given \(f(x)=2 x^{3}-11 x^{2}+7 x-5,\) use the Remainder Theorem to find \(f(4)\)

Use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range. \(f(x)=2 x-x^{2}-2\)

In Exercises \(27-34,\) find the zeros for each polynomial function and give the multiplicity for each zero. State whether the graph crosses the \(x\) -axis, or touches the \(x\) -axis and turns around, at each zero. $$f(x)=x^{3}+5 x^{2}-9 x-45$$

Use the four-step procedure for solving variation problems given on page 356 to solve. An object's weight on the moon, \(M,\) varies directly as its weight on Earth, \(E .\) A person who weighs 55 kilograms on Earth weights 8.8 kilograms on the moon. What is the moon weight of a person who weighs 90 kilograms on Earth?

In Exercises \(29-36,\) find an nth-degree polynomial function with real coefficients satisfying the given conditions. If you are using a graphing utility, use it to graph the function and verify the real zeros and the given function value. \(n=4 ;-2,-\frac{1}{2},\) and \(i\) are zeros; \(f(1)=18\)

Find all zeros of the polynomial function or solve the given polynomial equation. Use the Rational Zero Theorem, Descartes's Rule of Signs, and possibly the graph of the polynomial function shown by a graphing utility as an aid in obtaining the first zero or the first root. $$ f(x)=x^{4}-4 x^{3}-x^{2}+14 x+10 $$

Find the horizontal asymptote, if any, of the graph of each rational function. $$h(x)=\frac{15 x^{3}}{3 x^{2}+1}$$

Given \(f(x)=x^{3}-7 x^{2}+5 x-6,\) use the Remainder Theorem to find \(f(3)\).

Use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range. \(f(x)=6-4 x+x^{2}\)

In Exercises \(35-50\) a. Use the Leading Coefficient Test to determine the graphs end behavior. b. Find \(x\) -intercepts by setting \(f(x)=0\) and solving the resulting polynomial equation. State whether the graph crosses the \(x\)-axis, or touches the \(x\)-axis and turns around, at each intercept. c. Find the \(y\) -intercept by setting \(x\) equal to 0 and computing \(f(0)\) d. Determine whether the graph has \(y\) -axis symmetry, origin symmetry, or neither. e. If necessary, find a few additional points and graph the function. Use the fact that the maximum number of turning points of the graph is \(n-1\) to check whether it is drawn correctly. $$f(x)=x^{3}+2 x^{2}-x-2$$

In Exercises \(29-36,\) find an nth-degree polynomial function with real coefficients satisfying the given conditions. If you are using a graphing utility, use it to graph the function and verify the real zeros and the given function value. \(n=4 ;-2,5,\) and \(3+2 i\) are zeros; \(f(1)=-96\)

Find all zeros of the polynomial function or solve the given polynomial equation. Use the Rational Zero Theorem, Descartes's Rule of Signs, and possibly the graph of the polynomial function shown by a graphing utility as an aid in obtaining the first zero or the first root. $$ x^{4}-3 x^{3}-20 x^{2}-24 x-8=0 $$

Find the horizontal asymptote, if any, of the graph of each rational function. $$f(x)=\frac{-2 x+1}{3 x+5}$$