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Chapter Five: Polynomials and Polynomial Functions
Transcript of Chapter Five: Polynomials and Polynomial Functions
Polynomials and Polynomial Functions Chapter 5.5
Remainder and Factor Theorems Chapter 5.6
Find Rational Zeros Chapter 5.7
Fundamental Theorem of Algebra Chapter 5.9
Writing Polynonmial Functions definitions: Degree: the highest power that appears
Leading Coefficient: coefficient of the highest ordered term Concepts:
Zero: k is a zero
Factor: 'x-k' is a factor
Solution: K is a solution of 'F(x)=0'
X-intercept: If 'K' is a real number, 'K' is an X-intercept of the graph. The graph passes through '(k,0)' Turning Points: Local Maximum: on the Y-coordinate; it is the highest point of the graph
Local Minimum: on the Y-coordinate; it is the lowest point of the graph STeps: Step 1: Plot the intercepts
*(-3,0) and (2,0), because -3 and 2 are zeros of 'F'
Step 2: Plot the points between and after/before the X-intercepts Step 3: Find the end behavior; this one is positive
Step 4: Draw the graph, making it go through the points and with the right end behavior
Step 5: Find the local minimum/maximum; (-3,0)/(2,0) and (-2,8/3) Factor Theorem A polynomial f(x) has a factor (x - k) if and only if f(k) = 0. Write a cubic polynomial given the following points: Zeros:
X = -3
X = -2
X = -5 -15 = a(0+3)(0-2)(0-5) -15 =a (3)(-2)(-5) -15 = 30a divide by 30 on each side a = -1/2 The final remainder of the equation being divided (using synthetic division) equals zero. This follows the factor theorem because (x - k) = (x + 3) which means that (k = -3). Because the remainder is zero, the factor theorem is proven. Y = -1/2(X+3)(X-2)(X-5) Arielle Limberis
Rachel Inderhees Finite Differences - When the x-values in a data set are equally spaced, the differences of consecutive y-values are called finite differences. Properties of Finite Differences 1. If a polynomial function f(x) has degree n, then the nth-order differences of function values for equally-spaced x-values are nonzero and constant.
2. Conversely, if the nth-order differences of equally-spaced data are nonzero and constant, then the data can be represented by a polynomial function of degree n. Definitions (-3,0) (2,0) (5,0) (0,-15) Remainder Theorem If a polynomial f(x) is divided by (x - k), then the remainder is r = f(k). Suggested Review: Page 404-406: 25-39 odds 6 x - 5 Long Division Example Theorem: If f(x) is a polynomial of degree (n) where (n > 0), then the equation f(x) = 0 has at least one solution in the set of complex numbers.
Corollary: If f(x) is a polynomial of degree n where (n > 0), then the equation f(x) = 0 has exactly (n) solutions provided each solution repeated twice is counted as 2 solutions, each solution repeated three times is counted as 3 solutions, and so on. To find the number of zeros and/or solutions, look to the degree of the polynomial.
ie. has a degree of 3 and will therefore have three answers has a degree of 7 and therefore will have seven answers note: answers can be imaginaries, provided that when they are multiplied together the (i) is cancelled out. The Rational Zero Theorem If f(x) = has integer coefficients, then
every rational zero of f has the following form. Examples 1. Step to Finding the Rational Zeros: 1. Find the factors of the constant term. (term without the variable) 2. Find the factors of the leading coefficient. (number before the value with the highest degree) 3. Find the possible rational zeros. (factors of constant term/factors of leading coefficient) 4. Simplify the values from your list of possible zeros. Factors of constant term: ±1, ±2, ±3, ±4, ±6, ±12 Factors of the leading coefficient: ±1 Possible rational zeros: ± 1/1 , ±2/1, ±3/1, ±4/1, ±6/1, ±12/1 Simplified list: ±1, ±2, ±3, ±4, ±6, ±12 2. Rational Zero Theorem Limits: The Rational Zero Theorem only lists possible zeros. You have to test
the vaules from the list of possible zeros to find the actual zeros. 1 -8 11 20
1 -7 4 24 1 -7 4 1 Possible Zeros: ±1, ±2, ±4, ±5, ±10, ±20 Testing x = 1 Testing x = -1 -1 1 -8 11 20 -1 9 -20 1 -9 20 0 1 is not a zero. -1 is a zero. Chapter 5.8
Analyze Graphs of Polynomial Functions To get this equation, find the zeros from the points you were given, and plug them into the polynomial function:
Y = a(X+_)(X+_)(X+_)