1. Sketching Graphs of Polynomials 9 November 2010

2. Basic Info about Polynomials They are continuous 1 smooth line No breaks, jumps, or discontinuities

3. Key Features of Polynomials, Review a n x n + a n-1 x n-1 + … + a 2 x 2 + a 1 n 1 + a 0 ConstantLeading Coefficient (a n ) Degree (n, greatest exponent in polynomial)

4. Key Features of Polynomials, Review The constant is the y-intercept y = -4x 3 + 6x 2 – x + 3 y-intercept = 3

5. Finding the Roots for Polynomials Review Given a cubic, then you need: 1 root Synthetic division once Factor once Given a quartic, then you need: 2 roots Synthetic division twice Factor once

6. Finding the Roots for Polynomials Review, cont. For Cubic Equations: Step 1: Convert the root into a linear factor (if necessary) Step 2: Use synthetic division to solve for a second factor. (It will probably be a quadratic equation.) Step 3: Factor the quadratic equation. Step 4: Solve for x.

7. Finding the Roots for Polynomials Review, cont. For Quartic Equations: Step 1: Convert both roots into linear factors (if necessary) Step 2: Use synthetic division with one of the linear factors. Step 3: Use synthetic division on the quotient from step 2 with the other linear factor. Step 4: Factor the result of step 3. Step 5: Solve for x.

8. Solving for the Roots of Polynomials Practice Complete problems 1 – 4 on the “Solving for the Roots of Polynomials Review” handout.

9. Steps for Sketching Graphs of Polynomials Step 1: Identify the y-intercept Step 2: Classify the graph as either even or odd Step 3: Classify the leading coefficient as either positive or negative

10. Steps for Sketching Graphs of Polynomials, cont. Step 4: Solve for the roots of the polynomial Step 5: Complete a multiplicity table for the equation Step 6: Graph the roots and the y- intercept

11. Steps for Sketching Graphs of Polynomials, cont. Step 7: Connect the points. Pay attention to multiplicity (cross x-axis vs. touch x-axis)!!! Step 8: Sketch the end behavior (even vs. odd and positive vs. negative leading coefficient)

12. Example x = -4 is one of the roots of y = x 3 – 13x + 12 Step 1: y-intercept = Step 2: Even or Odd? Step 3: Leading Coefficient?

13. Example, cont. x = -4, y = x 3 – 13x + 12 Step 4: Solve for roots

14. Example, cont. x = -4, y = x 3 – 13x + 12 Step 5: Complete a multiplicity table RootLinear FactorMultiplicityX-Axis -4x + 41 or oddCross 1x – 11 or oddCross 3x – 31 or oddCross

15. Your Turn: Sketch the graphs of problems 1 – 4 on “Solving for the Roots of Polynomials Review” Handout

16. Homework Complete the following problems for homework: 1. x = −4 is a root of y = x 3 + 3x 2 − 6x − 8. Sketch the graph of y = x 3 + 3x 2 − 6x − 8. 2. x = −2 and x = −1 are roots of y = x 4 − 5x 2 + 4. Sketch the graph of y = x 4 − 5x 2 + 4.