Stewart Precalc 6e Chapter 3 Review

1 Quadratic Function - Standard Form · Level 2

A quadratic function is given. (a) Express the function in standard form. (b) Graph the function. \(f(x) = x^2 + 4 x + 1\).

2 Quadratic Function - Standard Form · Level 2

A quadratic function is given. (a) Express the function in standard form. (b) Graph the function. \(f(x) = -2 x^2 + 12 x + 12\).

3 Quadratic Function - Standard Form · Level 2

A quadratic function is given. (a) Express the function in standard form. (b) Graph the function. \(g(x) = 1 + 8 x - x^2\).

4 Quadratic Function - Standard Form · Level 2

A quadratic function is given. (a) Express the function in standard form. (b) Graph the function. \(g(x) = 6 x - 3 x^2\).

5 Quadratic Function - Maximum/Minimum · Level 2

Find the maximum or minimum value of the quadratic function \(f(x) = 2 x^2 + 4 x - 5\).

6 Quadratic Function - Maximum/Minimum · Level 2

Find the maximum or minimum value of the quadratic function \(g(x) = 1 - x - x^2\).

7 Quadratic Application - Projectile Motion · Level 3

A stone is thrown upward from the top of a building. Its height (in feet) above the ground after \(t\) seconds is given by the function \(h(t) = -16 t^2 + 48 t + 32\). What maximum height does the stone reach?

8 Quadratic Application - Profit Maximization · Level 3

The profit \(P\) (in dollars) generated by selling \(x\) units of a certain commodity is given by the function \(P(x) = -1500 + 12 x - 0.004 x^2\). What is the maximum profit, and how many units must be sold to generate it?

9 Polynomial Graph Transformation · Level 3

Graph the polynomial by transforming an appropriate graph of the form \(y = x^n\). Show clearly all \(x\)- and \(y\)-intercepts. \(P(x) = -x^3 + 64\).

10 Polynomial Graph Transformation · Level 3

Graph the polynomial by transforming an appropriate graph of the form \(y = x^n\). Show clearly all \(x\)- and \(y\)-intercepts. \(P(x) = 2 x^3 - 16\).

11 Polynomial Graph Transformation · Level 3

Graph the polynomial by transforming an appropriate graph of the form \(y = x^n\). Show clearly all \(x\)- and \(y\)-intercepts. \(P(x) = 2 (x + 1)^4 - 32\).

12 Polynomial Graph Transformation · Level 3

Graph the polynomial by transforming an appropriate graph of the form \(y = x^n\). Show clearly all \(x\)- and \(y\)-intercepts. \(P(x) = 81 - (x - 3)^4\).

13 Polynomial Graph Transformation · Level 3

Graph the polynomial by transforming an appropriate graph of the form \(y = x^n\). Show clearly all \(x\)- and \(y\)-intercepts. \(P(x) = 32 + (x - 1)^5\).

14 Polynomial Graph Transformation · Level 3

Graph the polynomial by transforming an appropriate graph of the form \(y = x^n\). Show clearly all \(x\)- and \(y\)-intercepts. \(P(x) = -3 (x + 2)^5 + 96\).

15 Remainder Theorem · Level 2

Find the indicated value of the polynomial using the Remainder Theorem. \(Q(x) = x^4 + 4 x^3 + 7 x^2 + 10 x + 15\); find \(Q(-3)\)

16 Factor Theorem · Level 2

Show that \(\dfrac{1}{2}\) is a zero of the polynomial \(P(x) = 2 x^4 + x^3 - 5 x^2 + 10 x - 4\)

17 Factor Theorem · Level 2

Use the Factor Theorem to show that \(x + 4\) is a factor of the polynomial \(P(x) = x^5 + 4 x^4 - 7 x^3 - 23 x^2 + 23 x + 12\)

18 Remainder Theorem · Level 2

What is the remainder when the polynomial \(P(x) = x^{500} + 6 x^{201} - x^2 - 2 x + 4\) is divided by \(x - 1\)?

19 Remainder Theorem · Level 2

What is the remainder when \(x^{101} - x^4 + 2\) is divided by \(x + 1\)?

20 Rational Zeros and Descartes' Rule · Level 2

A polynomial \(P\) is given. (a) List all possible rational zeros (without testing to see whether they actually are zeros). (b) Determine the possible number of positive and negative real zeros using Descartes' Rule of Signs. \(P(x) = x^5 - 6 x^3 - x^2 + 2 x + 18\)

21 Rational Zeros and Descartes' Rule · Level 2

A polynomial \(P\) is given. (a) List all possible rational zeros (without testing to see whether they actually are zeros). (b) Determine the possible number of positive and negative real zeros using Descartes' Rule of Signs. \(P(x) = 6 x^4 + 3 x^3 + x^2 + 3 x + 4\)

22 Real Zeros and Graph · Level 2

A polynomial \(P\) is given. (a) Find all real zeros of \(P\), and state their multiplicities. (b) Sketch the graph of \(P\). \(P(x) = x^3 - 16 x\)

23 Real Zeros and Graph · Level 2

A polynomial \(P\) is given. (a) Find all real zeros of \(P\), and state their multiplicities. (b) Sketch the graph of \(P\). \(P(x) = x^3 - 3 x^2 - 4 x\)

24 Real Zeros and Graph · Level 2

A polynomial \(P\) is given. (a) Find all real zeros of \(P\), and state their multiplicities. (b) Sketch the graph of \(P\). \(P(x) = x^4 + x^3 - 2 x^2\)

25 Real Zeros and Graph · Level 2

A polynomial \(P\) is given. (a) Find all real zeros of \(P\), and state their multiplicities. (b) Sketch the graph of \(P\). \(P(x) = x^4 - 5 x^2 + 4\)

26 Real Zeros and Graph · Level 3

A polynomial \(P\) is given. (a) Find all real zeros of \(P\), and state their multiplicities. (b) Sketch the graph of \(P\). \(P(x) = x^4 - 2 x^3 - 7 x^2 + 8 x + 12\)

27 Real Zeros and Graph · Level 3

A polynomial \(P\) is given. (a) Find all real zeros of \(P\), and state their multiplicities. (b) Sketch the graph of \(P\). \(P(x) = x^4 - 2 x^3 - 2 x^2 + 8 x - 8\)

28 Real Zeros and Graph · Level 3

A polynomial \(P\) is given. (a) Find all real zeros of \(P\), and state their multiplicities. (b) Sketch the graph of \(P\). \(P(x) = 2 x^4 + x^3 + 2 x^2 - 3 x - 2\)

29 Real Zeros and Graph · Level 4

A polynomial \(P\) is given. (a) Find all real zeros of \(P\), and state their multiplicities. (b) Sketch the graph of \(P\). \(P(x) = 9 x^5 - 21 x^4 + 10 x^3 + 6 x^2 - 3 x - 1\)

30 Complex Numbers · Level 1

Evaluate the expression and write in the form \(a + b i\). \((2 - 3 i) + (1 + 4 i)\)

31 Complex Numbers · Level 1

Evaluate the expression and write in the form \(a + b i\). \((3 - 6 i) - (6 - 4 i)\)

32 Complex Numbers · Level 1

Evaluate the expression and write in the form \(a + b i\). \((2 + i)(3 - 2 i)\)

33 Complex Numbers · Level 1

Evaluate the expression and write in the form \(a + b i\). \(4 i \left(2 - \dfrac{1}{2} i\right)\)

34 Complex Numbers · Level 2

Evaluate the expression and write in the form \(a + b i\). \(\dfrac{4 + 2 i}{2 - i}\)

35 Complex Numbers · Level 2

Evaluate the expression and write in the form \(a + b i\). \(\dfrac{8 + 3 i}{4 + 3 i}\)

36 Complex Numbers · Level 2

Evaluate the expression and write in the form \(a + b i\). \((1 + i)^3\)

37 Complex Numbers · Level 2

Evaluate the expression and write in the form \(a + b i\). \((1 - \sqrt{-1})(1 + \sqrt{-1})\)

38 Complex Numbers · Level 2

Evaluate the expression and write in the form \(a + b i\). \(\sqrt{-10} \cdot \sqrt{-40}\)

39 Polynomial Construction · Level 3

Find a polynomial of degree 3 with constant coefficient 12 and zeros \(-\dfrac{1}{2}\), \(2\), and \(3\).

40 Polynomial Construction · Level 3

Find a polynomial of degree 4 that has integer coefficients and zeros \(3 i\) and \(4\), with \(4\) a double zero.

41 Polynomial Construction · Level 3

Does there exist a polynomial of degree 4 with integer coefficients that has zeros \(i\), \(2 i\), \(3 i\), and \(4 i\)? If so, find it. If not, explain why.

42 Polynomial Reasoning · Level 3

Prove that the equation \(3 x^4 + 5 x^2 + 2 = 0\) has no real root.

43 Finding All Zeros · Level 3

\( P(x) = x^3 - 3 x^2 - 13 x + 15 \)

44 Finding All Zeros · Level 3

\( P(x) = 2 x^3 + 5 x^2 - 6 x - 9 \)

45 Finding All Zeros · Level 3

\( P(x) = x^4 + 6 x^3 + 17 x^2 + 28 x + 20 \)

46 Finding All Zeros · Level 3

\( P(x) = x^4 + 7 x^3 + 9 x^2 - 17 x - 20 \)

47 Finding All Zeros · Level 4

\( P(x) = x^5 - 3 x^4 - x^3 + 11 x^2 - 12 x + 4 \)

48 Finding All Zeros · Level 2

\( P(x) = x^4 - 81 \)

49 Finding All Zeros · Level 3

\( P(x) = x^6 - 64 \)

50 Finding All Zeros · Level 3

\( P(x) = 18 x^3 + 3 x^2 - 4 x - 1 \)

51 Finding All Zeros · Level 3

\( P(x) = 6 x^4 - 18 x^3 + 6 x^2 - 30 x + 36 \)

52 Finding All Zeros · Level 2

\( P(x) = x^4 + 15 x^2 + 54 \)

53 Graphing to Solve Equations · Level 2

Use a graphing device to find all real solutions of the equation. \(2 x^2 = 5 x + 3\)

54 Graphing to Solve Equations · Level 2

Use a graphing device to find all real solutions of the equation. \(x^3 + x^2 - 14 x - 24 = 0\)

55 Graphing to Solve Equations · Level 3

Use a graphing device to find all real solutions of the equation. \(x^4 - 3 x^3 - 3 x^2 - 9 x - 2 = 0\)

56 Graphing to Solve Equations · Level 3

Use a graphing device to find all real solutions of the equation. \(x^5 = x + 3\)

57 Factoring with Complex · Level 3

A polynomial function \(P\) is given. Find all the real zeros of \(P\), and factor \(P\) completely into linear and irreducible quadratic factors with real coefficients. \(P(x) = x^3 - 2 x - 4\)

58 Factoring with Complex · Level 2

A polynomial function \(P\) is given. Find all the real zeros of \(P\), and factor \(P\) completely into linear and irreducible quadratic factors with real coefficients. \(P(x) = x^4 + 3 x^2 - 4\)

59 Rational Function Graphing · Level 2

Graph the rational function. Show clearly all \(x\)- and \(y\)-intercepts and asymptotes. \(r(x) = \dfrac{3 x - 12}{x + 1}\)

60 Rational Function Graphing · Level 2

Graph the rational function. Show clearly all \(x\)- and \(y\)-intercepts and asymptotes. \(r(x) = \dfrac{1}{(x + 2)^2}\)

61 Rational Function Graphing · Level 3

Graph the rational function. Show clearly all \(x\)- and \(y\)-intercepts and asymptotes. \(r(x) = \dfrac{x - 2}{x^2 - 2 x - 8}\)

62 Rational Function Graphing · Level 3

Graph the rational function. Show clearly all \(x\)- and \(y\)-intercepts and asymptotes. \(r(x) = \dfrac{2 x^2 - 6 x - 7}{x - 4}\)

63 Rational Function Graphing · Level 2

Graph the rational function. Show clearly all \(x\)- and \(y\)-intercepts and asymptotes. \(r(x) = \dfrac{x^2 - 9}{2 x^2 + 1}\)

64 Rational Function Graphing · Level 3

Graph the rational function. Show clearly all \(x\)- and \(y\)-intercepts and asymptotes. \(r(x) = \dfrac{x^3 + 27}{x + 4}\)

65 Rational Function Asymptotes · Level 2

Use a graphing device to analyze the graph of the rational function. Find all \(x\)- and \(y\)-intercepts and all vertical, horizontal, and slant asymptotes. If the function has no horizontal or slant asymptote, find a polynomial that has the same end behavior as the rational function. \(r(x) = \dfrac{x - 3}{2 x + 6}\)

66 Rational Function Asymptotes · Level 2

Use a graphing device to analyze the graph of the rational function. Find all \(x\)- and \(y\)-intercepts and all vertical, horizontal, and slant asymptotes. If the function has no horizontal or slant asymptote, find a polynomial that has the same end behavior as the rational function. \(r(x) = \dfrac{2 x - 7}{x^2 + 9}\)

67 Rational Function Asymptotes · Level 3

Use a graphing device to analyze the graph of the rational function. Find all \(x\)- and \(y\)-intercepts and all vertical, horizontal, and slant asymptotes. \(r(x) = \dfrac{x^3 + 8}{x^2 - x - 2}\)

68 Rational Function Asymptotes · Level 3

Use a graphing device to analyze the graph of the rational function. Find all \(x\)- and \(y\)-intercepts and all vertical, horizontal, and slant asymptotes. If the function has no horizontal or slant asymptote, find a polynomial that has the same end behavior as the rational function. \(r(x) = \dfrac{2 x^3 - x^2}{x + 1}\)

69 Polynomial Intersections · Level 4

Find the coordinates of all points of intersection of the graphs of \(y = x^4 + x^2 + 24 x\) and \(y = 6 x^3 + 20\)

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