2.2 BEGINS: POLYNOMIAL
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1 CHAPTER 2.2 HIGHER DEGREE POLY S 2.2 BEGINS: POLYNOMIAL Graphs of Polynomial Functions Polynomial functions are continuous. What this means to us is that the graphs of polynomial functions have no breaks, holes, or gaps. Also polynomial are considered smooth! 1
2 CHARACTERISTICS OF POLYNOMIAL FUNCTIONS. Graphs of Polynomial Functions Polynomial functions have another characteristic. They have: Extrema: The multiple minimums and maximums of a function. Global = relative. IS THIS A POLYNOMIAL? ESTIMATE THE EXTREMA IF YOU CAN!!! 2
3 CONTINUOUS?? ESTIMATE THE EXTREMA IF YOU CAN!!! POLYNOMIAL?? IF SO, ESTIMATE THE EXTREMA IF YOU CAN!!! 3
4 DEGREES OF A POLYNOMIAL This is the highest exponent of a variable This would be a 111 th degree polynomial. If you have, you would added the exponents together. 7 th degree poly. We will cover this later in the book. LEADING COEFFICIENT TEST As x moves without bound to the left or to the right, the graph of the polynomial function f(x) =... eventually rises or falls in the following manner: 1. When n is odd: a. If the leading coefficient is positive, the graph falls to the left and rises to the right. b. If the leading coefficient is negative, the graph rises to the left and falls to the right. 2. When n is even: a. If the leading coefficient is positive, the graph rises to the left and right. b. If the leading coefficient is negative, the graph falls to the left and right. 4
5 A TABLE TO REMEMBER BY APPLICATION OF THE LEADING COEFFICIENT TEST What is the degree of the polynomial? 5
6 EXAMPLE #2 What is the degree of the polynomial? ZEROS OF POLYNOMIAL FUNCTIONS Let f be a polynomial function of degree n. The function f has at most n real zeros. The graph of f has at most n - 1 relative extrema. Example: This has up to extrema! How many times The graph crosses or touches the x-axis. 6
7 AT MOST HOW MANY ZEROS AND EXTREMA? 1) 2) 3) OTHER VOCABULARY Repeated zero If, 1 is a factor of a polynomial, then is a repeated zero. Multiplicity The number of times a zero is repeated. Example: We have a repeated zero of 2, and it s multiplicity is 3. 7
8 WHAT DOES THIS ALL MEAN? If a polynomial function f has a repeated zero x = 3 with multiplicity 4, the graph of f touches the x-axis at x = 3. (multiplicity is even, it touches) If f has a repeated zero x = 4 with multiplicity 3, the graph of f crosses the x- axis at x = 4. (multiplicity is odd, it crosses). [Note: Sometimes there is a little wiggle in the graph for this situation] ZEROS OF POLYNOMIAL FUNCTIONS Let f be a polynomial function and let a be a real number. Four equivalent statements about the real zeros of f: 1) is a zero of the function 2) is a solution of the polynomial equation 0 3) is a factor of the polynomial 4), 0 is an x-intercept of the graph of 8
9 EXAMPLE 3: FIND THE X INTERCEPTS Find the x-intercept of the graph of 1. Grouping method: 1. Note that in the above example, 1 is a repeated zero. In general, a factor, 1, yields a repeated zero of multiplicity k. If k is odd, the graph crosses the x-axis at x = a. If k is even, the graph only touches the x-axis at x = a. YOU TRY THE GROUPING METHOD 1) 9
10 EXAMPLE 4: CAN YOU SKETCH THE GRAPH? 2 3 Odd or even value of n? If the leading coefficient is positive, the graph rises to the left and right. Find it s zeros or intercepts. Stumped? What if I multiply it all by 4? What does this kind of look like? What if I subs. a for? Factor it now! Put back the. Now solve for each. EXAMPLE 4 CONTINUED With all of the information, we can surely graph to some degree of certainty. Rises to the left, and right. No multiplicity/repeated zeros. 10
11 FIND THE ZEROS: YOU TRY THE FACTORING METHOD 1) EXAMPLE #5 Sketch the graph of Since the x-intercepts are, and,. 2. The graph will go to right and to the left. 3. Additional points on the graph are 11
12 FINDING ZEROS #6,7 FIND A POSSIBLE POLY WITH GIVEN ZEROES. EXAMPLE 8 Find the polynomial function with the following zeros.,,, Why do you think they call this a possible polynomial? 12
13 FIND THE POLYNOMIAL WITH THE POSSIBLE ZEROS (OYO) 1),, FIND THE POLYNOMIAL WITH THE POSSIBLE ZEROS (OYO) 1), 13
14 EXAMPLE 9: HARDER ONE. LETS DO THIS! (DOUBLE FOIL) 0 H.W. P.109 #29-31,37,43-47ODD,49, 53, 55, 59,67,70,71,73, 81, 89, [CHALLENGE 109]. 14
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