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Basics of a Polynomial
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1. Basics of a Polynomial

2. Polynomial 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: Linear:

3. Zero or Root of a Polynomial A value for the independent variable (x) that makes the polynomial equal 0. Ex: What are the zeros of the following polynomial and how are they represented on the graph? These are the x-intercepts The zeros/roots are and

4. Coefficient of a Term A number that multiplies a variable or variable expression. Ex: In the polynomial below, what is the coefficient of x4?

5. Leading Coefficient of a Polynomial The coefficient of the term in a polynomial which contains the highest power of the variable. Ex: What is the leading coefficient of the polynomial below and how does it affect the graph? The leading coefficient acts as the “a” in our polynomial equation (similar to the “a” in ax2+bx+c). It will stretch or compress the graph and if it is negative, it will flip the graph. This stretches and flips the graph.

6. Factor of a Polynomial The expressions that multiply to get another expression. Ex: What are the factors of the following polynomial? Therefore, the factors are and

7. Example: Factors What do the factors of a polynomial tell us about the graph of the polynomial? The values that make each factor equal zero are the x-intercepts.

8. Example: y-intercept How can you find the y-intercept of each equation without a table or graph? 8 Remember you can still substitute 0 for x to find the y-intercept. In standard form, the term without an “x.” In factored form, the product of the numbers inside of the factors (w/o an “x”) and the leading coefficient.

9. Degree of a Polynomial Highest power of an independent variable in a polynomial equation. Ex: What are the degrees of the following polynomials? 7 5 The degree is the number of factors

10. Example: Degree What does the degree of a polynomial tell us about the graph of the polynomial? Degree = 4 Degree = 4 Degree = 3 The degree is the maximum number of roots. Even degrees have the same end behavior. Odd degrees have opposite end behavior.

11. Repeated Root of a Polynomial A value for x that makes more than one factor equal zero. Ex: What is the repeated root of the polynomial below? Therefore, the repeated root is

12. Example: Repeated Roots What does the degree of a polynomial tell us about the graph of the polynomial? NOTE: If x is outside of the parentheses in factored form, 0 is an x-intercept. An even repeated root “bounces” off the x-axis. An odd repeated root “twists” through the x-axis.

13. Example: Could Be v Must Be x-intercepts: -4, 1, and 5 Factors: (x + 4), (x – 1), and (x – 5) Count the roots Degree: Even 5,6 4 1,2,3 Minimum Degree: 6 Orientation: Positive (opens up) “a”: Positive y-intercept: ~-5 Even Repeated Root Odd Repeated Root

14. 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

15. 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

16. 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

17. 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

18. 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) -2 3 5 x-intercepts: Zero-Product Property Negative Orientation (start “up” then go “down”)

19. Example: Polynomial Equations to Graphs Without a calculator describe the general shape of: 4 3 – 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)

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

21. Complete Graph Pay Attention to end behavior On graph paper: Plot key points Scale your axes appropriately Plot points accurately When a problem says graph an equation or draw a graph: