Chapter 3 Polynomial Functions

Transcription

1 Trig / Coll. Alg. Name: Chapter 3 Polynomial Functions 3.1 Quadratic Functions (not on this test) For each parabola, give the vertex, intercepts (x- and y-), axis of symmetry, and sketch the graph. 1. f x x x ( ) = 4 5. f x ( x ) ( ) = Write the equation (in vertex form) of the parabola having vertex (3, ) that contains the point (5, 4).

2 3.b Zeros, Multiplicity and Graphing Polynomial Functions Review: Identify the left- and right-hand behavior of each function. A) f(x) = x 3 + 4x B) f(x) = x 4 5x + 4 left: right: maximum number of turns: left: right: maximum number of turns: Determine the intervals over which the function is increasing, decreasing and/or constant: C) Increasing: Decreasing: Constant: ***Notice that the transformations we have worked with in the past for quadratic and absolute value functions, remain the same for power functions. Review: Given f(x) is a power function: a) f(x) + a results in a shift of units b) g(x) = f(x + b) results in a shift of units c) g(x) = f(x) results in a over the Features of Graphs of Polynomial Functions: Graph is continuous no breaks only smooth turns. A graph of a polynomial function has at most turns where n is the of the polynomial. A polynomial function of degree n has zeros, although not all of the zeros must be. Some zeros may be numbers.

3 I. Use your Graphing Calculator to graph: f ( x) = x 6x + 9x What are the x-intercepts of the graph? These are also called the of the function or the or of the equation (when f ( x ) = 0 ). Notice that the graph through the x-axis at but only the x-axis at. Algebraically find the zeros of the function and compare these to your graph. {Find the values of x for which f ( x ) = 0 } Multiplicity: The graph of a polynomial function will CROSS an x-intercept (zero) having multiplicity, and only TOUCH an x-intercept (zero) that has multiplicity. II. Find all the real zeros of the polynomial function, determine the multiplicity of each, and determine whether the graph crosses or touches at each zero. 1. h t t t ( ) = g( x) = x + 6x x p( r) = r + r 6r

4 4. a. Find a polynomial function that has the given zeros: 0,, 1 b. Find another polynomial function that has the same zeros. III. Graph the polynomial function WITHOUT using your graphing calculator. Steps: 5. Use the degree and leading coefficient to determine the general shape and end behavior of the graph. Determine the zeros of the polynomial and their multiplicities. Plot at least one point between each zero and draw a continuous curve through the points. f ( x) x x 4 = 6. f ( x) = 4x + 4x + 15x h( x) = x ( x 4)

5 3.3 Synthetic & Long Division - Factoring and Zeros Review: Find all the real zeros of the polynomial function and determine the multiplicity of each zero. A. f ( x) x x 0x 4 = B. g x x x ( ) = C = + (Hint: factor out ½ ) h( x) x x D. Find a polynomial of degree that has a zero of. I. Use synthetic division to divide ( 3x 3 17x 15x 5) remainder. + by ( x 5). Give the quotient and Is ( x 5) a factor of the polynomial?. Use synthetic division to divide ( x 6 4x 4 3x ) + + by ( x + 1). Give the quotient and remainder. Is ( x + 1) a factor of the polynomial?

6 f x is divided by ( x k ) Remainder Theorem: If a polynomial ( ) the remainder is f ( k )., 3. Determine the remainder without dividing: ( 3x 3 8x 5x 7) ( x ) ***Note: synthetic division = synthetic substitution Use synthetic substitution to evaluate: 4. a. f ( x) = x x 10x + 5 for x = 3 5. a. Find f () for f x x x 3 ( ) = b. Is ( x 3) a factor of f ( x )? c. Name a point on the graph of f ( x ). b. Is ( x ) a factor of f ( x )? c. Name a point on the graph of f ( x ). Factor Theorem: A polynomial ( ) if and only if f ( k ) =0. f x has a factor ( x k )

7 6. Factor f ( x ) completely and find the remaining zeros of f ( x ) given that: f x x x x x 4 ( ) = + and ( x ) & ( 3) x + are factors of f ( x ) Synthetic division/substitution is easily used when the divisor is in the form ( x k ). If the divisor is not in this form, however, long division can be used instead. The steps for long division of polynomials are the same as the steps you used when you first learned how to do long division. It can be helpful to insert 0 s for missing powers just as you would for synthetic division. 7. Use long division to divide x 10x by ( 3) x Divide: ( x 9) ( x + 1)

8 3.4a Possible Rational Roots and Descartes Rule of Signs I. Possible Rational Roots : If a polynomial function f ( x ) has 1 or more rational roots, then the p root(s) will be in the form q ± where p = and q =. 1. List the possible rational roots of ( ) f x = x + x x +. Use your graphing calculator to find which, if any, of the possible rational roots are zeros of the function.. List the possible rational roots of 3 ( ) 3 6 f x = x + x. Use your graphing calculator to find which, if any, of the possible rational roots are zeros of the function. What can you conclude about the zeros of this function? Complex Zeros occur as. In other words, if is a zero of f ( x ), then must also be a zero of f ( x ). 3. Find all of the zeros of USE YOUR CALCULATOR! 4 ( ) f x = x x + x x given that 1+ 3i is a zero. DO NOT

9 II. Descartes Rule of Signs is a rule used to determine the possible number of + and zeros of a polynomial function. ***Sign Variation: when the sign on consecutive terms of the Possible # of positive zeros: polynomial changes from + to, or from to +. Possible # of negative zeros: 4. Use Descartes Rule of signs to find the possible number of positive and negative zeros of 7 4 f x = x + x + x x Positive: Negative: ( ) Use Descartes Rule of signs to find the possible number of positive and negative zeros of g x = x x + x Positive: Negative: ( ) 4 1 ( ) Given f x = x x x +, a.) List the possible rational zeros. b.) Use your graphing calculator to identify an actual rational zero, if possible. c.) Use synthetic division to verify and find all the remaining zeros. d.) Write the function in factored form.

10 3.5: Variations If y varies directly with x, or y is directly proportional to x, then y proportionality or constant of variation. ( k 0 ) = kx, where k is the constant of If y varies inversely with x, or y is inversely proportional to x, then proportionality. ( k 0 ) k y =, where k is the constant of x When a variable quantity y is proportional to the product of two or more other variables, we say that y varies jointly with these quantities. If y varies jointly with a and b, then y = kab. Combinations of direct, joint, and/or inverse variation may occur. This is usually referred to as combined variation. 1. The amount a spring will stretch, S, varies directly with the force (or weight), F, attached to the spring. If a spring stretches 1.8 inches with 60 pounds attached, how far will it stretch with 90 pounds attached?. On planet X, an object falls 13 feet in seconds. Knowing the distance it falls varies directly with the square of the time of fall, how long does it take an object to fall 75 feet? Round your answer to three decimal places. 3. The price per person of renting a bus varies inversely with the number of people renting the bus. It costs $17 per person if 37 people rent the bus. How much will it cost per person if 78 people rent the bus? 4. The wattage rating of an appliance, W, varies jointly as the square of the current, I, and the resistance, R. If the wattage is watts when the current is 0. ampere and the resistance is 50 ohms, find the wattage when the current is 0.4 ampere and the resistance is 100 ohms.