Polynomial Equations and Graphs

Polynomial Quadratics: An expression involving a sum of whole number powers multiplied by coefficients: anxn + … + a2x2 + a1x + a0 Ex: What are examples of polynomials that we have used frequently? Quadratics:

Polynomial Terms Polynomial: An expression involving a sum of whole number powers multiplied by coefficients: anxn + … + a2x2 + a1x + a0 Degree: Highest power of a variable. Factors: The expressions that multiply to get another expression: Since x2+3x+2=(x+1)(x+2), x+1 and x+2 are factors Zero or Root: A value for x that makes the polynomial 0. Coefficient: A number that multiplies a variable. Leading Coefficient: The coefficient of the term in a polynomial which contains the highest power of the variable. Repeated Root: A value for x that makes more than one factor equal zero. For instance -3 is a double root of (x+3)2(x-2)=(x+3)(x+3)(x-2).

Local Minimum/Local Maximum The lowest or highest point (listed as a coordinate) in a particular section of a graph. Local Maximum (1,5) (0,3) Local Minimum (5,-4) An Actual Minimum

Characteristics from a Graph Local Maximum Minimum Degree: 6 Degree: Even Factors: (x + 4), (x – 1), and (x – 5) x-intercepts: -4, 1, and 5 Minimum Orientation: Positive Local Minimum “a”: Positive y-intercept: ~-5 Repeated Root

Polynomial Equations to Graphs Roughly Sketch the general shape of: 1 2 3 -10 -7 12 Degree = 3 Opposite end behavior (odd) x-intercepts: Zero-Product Property

Polynomial Equations to Graphs Roughly Sketch the general shape of: 1 2 3 4 x-intercepts: Zero-Product Property Degree = 4 Identical end behavior (even) -6 -3 5 8

Polynomial Equations to Graphs Roughly Sketch the general shape of: – 1 2 3 4 5 x-intercepts: Zero-Product Property Degree = 5 Opposite end behavior (odd) Negative Orientation (start “up” then go “down”) -4 -2 6 10 15

Polynomial Equations to Graphs Roughly Sketch the general shape of: 2 2 1,2 3,4 2 Double Roots (bounce off the x-axis) Degree = 4 Identical end behavior (even) x-intercepts: Zero-Product Property -7 7

Polynomial Equations to Graphs Roughly Sketch the general shape of: 2 1,2 3 4 2 Double Roots (bounce off the x-axis) Degree = 4 Identical end behavior (even) x-intercepts: Zero-Product Property -2 3 5

Degree of a Polynomial A lot of the characteristics of a quadratic or cubic polynomial also hold for any polynomial. Yet, here is a summary of the new generalizations about the degree seen in the previous slides. The degree (the Highest power of a variable in a polynomial) determines the following: The maximum number of roots. End Behavior If the degree is even, the end behavior is identical (either “up” on both ends or “down” on both ends) If the degree is odd, the end behavior is opposite (either start “down” then go “up” OR start “up” then go “down”)

Polynomial Equations to Graphs Without a calculator describe the general shape of: 3 4 – 10 The value of the constant term determines the y-intercept. The sign of the leading coefficient determines the orientation. Whether the degree is even or odd determines the end behavior. AND The value of the degree determines the maximum number of roots. Orientation: Positive End Behavior: Identical (“up” on both ends) x-intercept(s): At most 4 roots. They can not be determined since it is not in factored. y-intercept: (0,-10)

Equation of a Polynomial to the Graph 2,3 5,6,7 1 4 – Triple Root Different end behavior (odd) Double Root Degree: 7 x-intercepts: -5, -1, 3, 6 (Zero Product Property) Orientation: Negative (since the degree is odd, start “up” then go “down”) y-intercept: