1. FACTORING & ANALYZING AND GRAPHING POLYNOMIALS

2. Analyzing To analyze a graph you must find: End behavior Max #of turns Number of real zeros(roots) Critical points Graph

3. End Behavior End behavior: This describes the far left and right of the graph and is determined by the coefficient and degree of the leading term. The Right side: if the leading coefficient is positive then the graph goes up to the right but if it's negative then it falls to the left. The Left side: if the degree of the leading term is even then the left remains the same as the right side but if the term is odd then the left is opposite.

4. Example y=4x³ – 3x : leading coefficient is 4x³ 4 is positive so the graph goes up to the right and the degree is 3 which is odd so the left is opposite of the right and falls on this side. END BEHAVIOR:  Right side: f(x)-->+∞ as x--> +∞  Left side: f(x)--> -∞ as x--> -∞

5. Max # of Turns and # of Real Roots Max Number of Turns: the amount of times the graph changes direction depends on the degree of the leading term. It is 1 less than that degree. Number of Real Roots: if polynomial is an odd degree it must cross the x-axis at least once. If it is even then it will cross the x-axis an even number of times

6. Derivatives Rules: the derivative of a constant with 1 term is 0. If you have a term such as bxª you find the derivative with this equation (a)(b)xª-1

7. Graph Just plug this into your calculator and when drawing follow these steps: Know how many turns you will have Start from the left and go up to the RELATIVE MAX then to the RELATIVE MIN—you might have to go back to the RELATIVE MAX if there are more than 2 turns. Don't forget that you cross the y-axis on the POINT OF INFLECTION

8. Now on to Factoring Polynomials

9. Synthetic Division Easiest way to explain would be to show Example: divide 3x³ – 4x² + 5x – 7 by x – 2 2 3 -4 5 -7 6 4 18 3 2 9 11 Always bring down first coefficient then multiply this number by the root and write the product under the next coefficient. Now add while writing the sums below the addition line. The numbers below addition line are quotient and remainder and the degree is always one less than the degree that you started with: 3x²+2x+9 R of 11

10. Synthetic Division cont. A non example would be: Divide x³ – 12 by x – 2 2 1 12 2 1 24 = 1x² + 24 This is a non example because there were no place holders added. Two place holders (0) were needed between the 1 and 12.

11. Finding Roots/Zeros To find the possible rational roots you use p/q where p=all factors of last term and q= all factors of leading coefficient. Example: list all possible rational zeros of each function 2x³+3x²-8x+3 p=±1, ±3 q=±1, ±2 so the possible zeros are: ±1, ±1/2, ±3, ±3/2 (used p/q one term at a time as if you were using FOIL method)

12. Finding Roots/Zeros cont. After you have found the possible rational roots you must try the possibilities until you find one with a remainder of 0. (with synthetic division) Example using the example from last slide: try 1 as a possible root for 2x³+3x²-8x+3 1 2 3 -8 3 2 5 -3 2 5 -3 0 new quotient: 2x²+5x-3 Use quadratic formula for other 2 roots: -3 & 1/2