Polynomial Functions

Polynomial Functions

1. Recognize and identify characteristics of Polynomial Functions

   • End Behaviour
     • Direction of arms
       • even degree polynomial has both arms in the same direction
       • odd degree polynomial has the arms in opposite directions
     • Direction of Right Arm
       • a_n > 0 right arm up
       • a_n < 0 right arm down
   • Turning Point
     • odd degree polynomial has an even number of turning points (n-1)
     • even degree polynomial has an odd number of turning points (n-1)
   • x-intercepts
     • Shape of the function
       • order of 2, parabolic shape
       • order of 1, linear shape
       • order of 3, cubic shape
     • Number of x-intercepts
       • even degree polynomial has between 0 and "n" x-intercepts
       • odd degree polynomial has between 1 and "n" x-intercepts

2. Apply algebraic procedures to simplify Polynomial Expressions

   • long division
   • Synthetic division
   • Remainder Theorem
     • When a polynomial P(x) is divided by x-b, P(b) = remainder
     • When a polynomial P(x) is divided by ax-b, P(b/a) = remainder
   • Factor Theorem
     • If P(x)is a polynomial, and (x-b)its divisor, then if P(b)=0, (x-b) is a factor
     • If P(x)is a polynomial, and (ax-b)its divisor, then if P(b)=0, (ax-b) is a factor

3. Solve polynomial equations and inequalities in a variety of ways – graphing by hand, technology, algebra, logic

   • Solve a polynomial by finding its roots
     • Solve the inequality by finding roots, graphing and stating the intervals
       • the values in the round brackets ( ... ) means that they do not include the value as represented by these inequalities < OR >
       • the values in the square brackets [ ... ] means that they do include the value as represented by these inequalities ≥ OR ≤

4. Make connections between Polynomial Functions and their graphs

   • Sketch a graph of a polynomial function from its characteristics
   • Determine the equation of the function from its graph
     • solve for (a) by substituting the y-intercept or a different point from the graph

5. Analyze, interpret, represent and communicate real world problems using Polynomial modeling techniques

   • create a scatterplot by plotting the points
     • Find a function to model the data using regression
       • this is for a cubic function y~ax^3 + bx^2 + cx + d

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