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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 2 Polynomial, Power, and Rational Functions
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 2.1 Linear and Quadratic Functions and Modeling
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Slide 2- 4 Quick Review
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Slide 2- 5 Quick Review Solutions
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Slide 2- 6 What you’ll learn about Polynomial Functions Linear Functions and Their Graphs Average Rate of Change Linear Correlation and Modeling Quadratic Functions and Their Graphs Applications of Quadratic Functions … and why Many business and economic problems are modeled by linear functions. Quadratic and higher degree polynomial functions are used to model some manufacturing applications.
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Slide 2- 7 Polynomial Function
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Slide 2- 8 Polynomial Functions of No and Low Degree NameFormDegree Zero Functionf(x) = 0Undefined Constant Functionf(x) = a (a ≠ 0)0 Linear Functionf(x)=ax + b (a ≠ 0)1 Quadratic Functionf(x)=ax 2 + bx + c (a ≠ 0)2
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Slide 2- 9 Example Finding an Equation of a Linear Function
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Slide 2- 10 Example Finding an Equation of a Linear Function
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Slide 2- 11 Average Rate of Change
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Slide 2- 12 Constant Rate of Change Theorem A function defined on all real numbers is a linear function if and only if it has a constant nonzero average rate of change between any two points on its graph.
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Slide 2- 13 Characterizing the Nature of a Linear Function Point of ViewCharacterization Verbalpolynomial of degree 1 Algebraic f(x) = mx + b (m ≠ 0) Graphicalslant line with slope m and y-intercept b Analyticalfunction with constant nonzero rate of change m: f is increasing if m > 0, decreasing if m < 0; initial value of the function = f(0) = b
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Slide 2- 14 Properties of the Correlation Coefficient, r 1. -1 ≤ r ≤ 1 2. When r > 0, there is a positive linear correlation. 3. When r < 0, there is a negative linear correlation. 4. When |r| ≈ 1, there is a strong linear correlation. 5. When |r| ≈ 0, there is weak or no linear correlation.
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Slide 2- 15 Linear Correlation
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Slide 2- 16 Regression Analysis 1. Enter and plot the data (scatter plot). 2. Find the regression model that fits the problem situation. 3. Superimpose the graph of the regression model on the scatter plot, and observe the fit. 4. Use the regression model to make the predictions called for in the problem.
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Slide 2- 17 Example Transforming the Squaring Function
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Slide 2- 18 Example Transforming the Squaring Function
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Slide 2- 19 The Graph of f(x)=ax 2
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Slide 2- 20 Vertex Form of a Quadratic Equation Any quadratic function f(x) = ax 2 + bx + c, a ≠ 0, can be written in the vertex form f(x) = a(x – h) 2 + k The graph of f is a parabola with vertex (h,k) and axis x = h, where h = -b/(2a) and k = c – ah 2. If a > 0, the parabola opens upward, and if a < 0, it opens downward.
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Slide 2- 21 Find the vertex and the line of symmetry of the graph y = (x – 1) 2 + 2 Domain Range (- , ) [2, ) Vertex (1,2) x = 1
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Slide 2- 22 Find the vertex and the line of symmetry of the graph y = -(x + 2) 2 - 3 Domain Range (- , ) (- ,-3] Vertex (-2,-3) x = -2
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Slide 2- 23 Let f(x) = x 2 + 2x + 4. (a) Write f in standard form. (b) Determine the vertex of f. (c) Is the vertex a maximum or a minimum? Explain f(x) = x 2 + 2x + 4 f(x) = (x + 1) 2 + 3 Vertex (-1,3) opens up (-1,3) is a minimum + 1- 1
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Slide 2- 24 Let f(x) = 2x 2 + 6x - 8. (a) Write f in standard form. (b) Determine the vertex of f. (c) Is the vertex a maximum or a minimum? Explain f(x) = 2(x + 3/2) 2 - 25/2 Vertex (-3/2,-25/2) opens up (-3/2,25/2) is a minimum + 9/4 - 9/2f(x) = 2(x 2 + 3x ) - 8
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Slide 2- 25 If we perform completing the square process on f(x) = ax 2 + bx + c and write it in standard form, we get
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Slide 2- 26 So the vertex is
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Slide 2- 27 To get the coordinates of the vertex of any quadratic function, simply use the vertex formula. If a > 0, the parabola open up and the vertex is a minimum. If a < 0, the parabola opens down and the parabola is a maximum. 1.5 Quadratic Functions
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Slide 2- 28 Example Finding the Vertex and Axis of a Quadratic Function
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Slide 2- 29 Example Finding the Vertex and Axis of a Quadratic Function
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Slide 2- 30 Characterizing the Nature of a Quadratic Function Point of View Characterization
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Slide 2- 31 Vertical Free-Fall Motion
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 2.2 Power Functions and Modeling
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Slide 2- 33 Quick Review
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Slide 2- 34 Quick Review Solutions
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Slide 2- 35 What you’ll learn about Power Functions and Variation Monomial Functions and Their Graphs Graphs of Power Functions Modeling with Power Functions … and why Power functions specify the proportional relationships of geometry, chemistry, and physics.
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Slide 2- 36 Power Function Any function that can be written in the form f(x) = k·x a, where k and a are nonzero constants, is a power function. The constant a is the power, and the k is the constant of variation, or constant of proportion. We say f(x) varies as the a th power of x, or f(x) is proportional to the a th power of x.
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Slide 2- 37 Example Analyzing Power Functions
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Slide 2- 38 Example Analyzing Power Functions
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Slide 2- 39 Monomial Function Any function that can be written as f(x) = k or f(x) = k·x n, where k is a constant and n is a positive integer, is a monomial function.
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Slide 2- 40 Example Graphing Monomial Functions
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Slide 2- 41 Example Graphing Monomial Functions
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Slide 2- 42 Graphs of Power Functions For any power function f(x) = k·x a, one of the following three things happens when x < 0. f is undefined for x < 0. f is an even function. f is an odd function.
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Slide 2- 43 Graphs of Power Functions
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 2.3 Polynomial Functions of Higher Degree with Modeling
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Slide 2- 45 Quick Review
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Slide 2- 46 Quick Review Solutions
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Slide 2- 47 What you’ll learn about Graphs of Polynomial Functions End Behavior of Polynomial Functions Zeros of Polynomial Functions Intermediate Value Theorem Modeling … and why These topics are important in modeling and can be used to provide approximations to more complicated functions, as you will see if you study calculus.
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Slide 2- 48 The Vocabulary of Polynomials
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Slide 2- 49 Example Graphing Transformations of Monomial Functions
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Slide 2- 50 Example Graphing Transformations of Monomial Functions
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Slide 2- 51 Cubic Functions
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Slide 2- 52 Quartic Function
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Slide 2- 53 Local Extrema and Zeros of Polynomial Functions A polynomial function of degree n has at most n – 1 local extrema and at most n zeros.
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Slide 2- 54 Leading Term Test for Polynomial End Behavior
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Slide 2- 55 Example Applying Polynomial Theory
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Slide 2- 56 Example Applying Polynomial Theory
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Slide 2- 57 Example Finding the Zeros of a Polynomial Function
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Slide 2- 58 Example Finding the Zeros of a Polynomial Function
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Slide 2- 59 Multiplicity of a Zero of a Polynomial Function
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Slide 2- 60 Example Sketching the Graph of a Factored Polynomial
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Slide 2- 61 Example Sketching the Graph of a Factored Polynomial
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Slide 2- 62 Intermediate Value Theorem If a and b are real numbers with a < b and if f is continuous on the interval [a,b], then f takes on every value between f(a) and f(b). In other words, if y 0 is between f(a) and f(b), then y 0 =f(c) for some number c in [a,b].
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 2.4 Real Zeros of Polynomial Functions
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Slide 2- 64 Quick Review
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Slide 2- 65 Quick Review Solutions
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Slide 2- 66 What you’ll learn about Long Division and the Division Algorithm Remainder and Factor Theorems Synthetic Division Rational Zeros Theorem Upper and Lower Bounds … and why These topics help identify and locate the real zeros of polynomial functions.
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Slide 2- 67 Division Algorithm for Polynomials
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Slide 2- 68 Example Using Polynomial Long Division
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Slide 2- 69 Example Using Polynomial Long Division
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Slide 2- 70 Remainder Theorem
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Slide 2- 71 Example Using the Remainder Theorem
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Slide 2- 72 Example Using the Remainder Theorem
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Slide 2- 73 Factor Theorem
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Slide 2- 74 Example Using Synthetic Division
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Slide 2- 75 Example Using Synthetic Division
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Slide 2- 76 Rational Zeros Theorem
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Slide 2- 77 Upper and Lower Bound Tests for Real Zeros
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Slide 2- 78 Example Finding the Real Zeros of a Polynomial Function
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Slide 2- 79 Example Finding the Real Zeros of a Polynomial Function
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Slide 2- 80 Example Finding the Real Zeros of a Polynomial Function
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Slide 2- 81 Example Finding the Real Zeros of a Polynomial Function
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 2.5 Complex Zeros and the Fundamental Theorem of Algebra
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Slide 2- 83 Quick Review
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Slide 2- 84 Quick Review Solutions
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Slide 2- 85 What you’ll learn about Two Major Theorems Complex Conjugate Zeros Factoring with Real Number Coefficients … and why These topics provide the complete story about the zeros and factors of polynomials with real number coefficients.
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Slide 2- 86 Fundamental Theorem of Algebra A polynomial function of degree n has n complex zeros (real and nonreal). Some of these zeros may be repeated.
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Slide 2- 87 Linear Factorization Theorem
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Slide 2- 88 Fundamental Polynomial Connections in the Complex Case The following statements about a polynomial function f are equivalent if k is a complex number: 1. x = k is a solution (or root) of the equation f(x) = 0 2. k is a zero of the function f. 3. x – k is a factor of f(x).
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Slide 2- 89 Example Exploring Fundamental Polynomial Connections
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Slide 2- 90 Example Exploring Fundamental Polynomial Connections
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Slide 2- 91 Complex Conjugate Zeros
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Slide 2- 92 Example Finding a Polynomial from Given Zeros
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Slide 2- 93 Example Finding a Polynomial from Given Zeros
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Slide 2- 94 Factors of a Polynomial with Real Coefficients Every polynomial function with real coefficients can be written as a product of linear factors and irreducible quadratic factors, each with real coefficients.
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Slide 2- 95 Example Factoring a Polynomial
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Slide 2- 96 Example Factoring a Polynomial
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 2.6 Graphs of Rational Functions
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Slide 2- 98 Quick Review
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Slide 2- 99 Quick Review Solutions
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Slide 2- 100 What you’ll learn about Rational Functions Transformations of the Reciprocal Function Limits and Asymptotes Analyzing Graphs of Rational Functions … and why Rational functions are used in calculus and in scientific applications such as inverse proportions.
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Slide 2- 101 Rational Functions
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Slide 2- 102 Example Finding the Domain of a Rational Function
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Slide 2- 103 Example Finding the Domain of a Rational Function
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Slide 2- 104 Graph a Rational Function
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Slide 2- 105 Graph a Rational Function
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Slide 2- 106 Example Finding Asymptotes of Rational Functions
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Slide 2- 107 Example Finding Asymptotes of Rational Functions
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Slide 2- 108 Example Graphing a Rational Function
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Slide 2- 109 Example Graphing a Rational Function
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 2.7 Solving Equations in One Variable
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Slide 2- 111 Quick Review
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Slide 2- 112 Quick Review Solutions
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Slide 2- 113 What you’ll learn about Solving Rational Equations Extraneous Solutions Applications … and why Applications involving rational functions as models often require that an equation involving fractions be solved.
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Slide 2- 114 Extraneous Solutions When we multiply or divide an equation by an expression containing variables, the resulting equation may have solutions that are not solutions of the original equation. These are extraneous solutions. For this reason we must check each solution of the resulting equation in the original equation.
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Slide 2- 115 Example Solving by Clearing Fractions
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Slide 2- 116 Example Solving by Clearing Fractions
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Slide 2- 117 Example Eliminating Extraneous Solutions
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Slide 2- 118 Example Eliminating Extraneous Solutions
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Slide 2- 119 Example Finding a Minimum Perimeter
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Slide 2- 120 Example Finding a Minimum Perimeter
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 2.8 Solving Inequalities in One Variable
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Slide 2- 122 Quick Review
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Slide 2- 123 Quick Review Solutions
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Slide 2- 124 What you’ll learn about Polynomial Inequalities Rational Inequalities Other Inequalities Applications … and why Designing containers as well as other types of applications often require that an inequality be solved.
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Slide 2- 125 Polynomial Inequalities
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Slide 2- 126 Example Finding where a Polynomial is Zero, Positive, or Negative
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Slide 2- 127 Example Finding where a Polynomial is Zero, Positive, or Negative -34 (-)(-) 2 (+)(-) 2 (+)(+) 2 negative positive
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Slide 2- 128 Example Solving a Polynomial Inequality Graphically
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Slide 2- 129 Example Solving a Polynomial Inequality Graphically
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Slide 2- 130 Example Creating a Sign Chart for a Rational Function
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Slide 2- 131 Example Creating a Sign Chart for a Rational Function -31 (-) (-)(-) negative positive (-) (+)(-) (+) (+)(+) (+) (+)(-) negative 0und.
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Slide 2- 132 Example Solving an Inequality Involving a Radical
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Slide 2- 133 Example Solving an Inequality Involving a Radical 2 (-)(+)(+)(+) undefined positivenegative 00
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Slide 2- 134 Chapter Test
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Slide 2- 135 Chapter Test
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Slide 2- 136 Chapter Test
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Slide 2- 137 Chapter Test Solutions
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Slide 2- 138 Chapter Test Solutions
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Slide 2- 139 Chapter Test Solutions
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