Zeros of a Polynomial AKA X-Intercepts; AKA Roots; AKA Solutions: The will tell you how many zeros the polynomial has.

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1 Example: Example:

2 Example: Zeros of a Polynomial AKA X-Intercepts; AKA Roots; AKA Solutions: The will tell you how many zeros the polynomial has. # of zeros = Multiplicity how many of the same factors occur. Even Multiplicity: graph touches and turns at the zero. Odd Multiplicity: graph crosses the zero.

3 Example: If g(5) = 0, what point is on the graph of g? The point is on the graph of g. (Type an ordered pair). The corresponding x-intercept of the graph of g is. (Type an integer or a fraction.) Example: Form a polynomial whose zeros and degree are given. Zeros: -4, multiplicity 1; -1, multiplicity 2; degree 3 The polynomial has integer coefficients and a leading coefficient of 1. Example:

4 Graphing Polynomials 1. Leading Coefficient Test 2. Possible Turning Points of a Polynomial Degree of the Polynomial 1 3. Behavior Near Zero (x- intercept) a) Substitute a zero for a factor that does not make f(x) = 0. b) Simplify c) The zero will take the behavior of the simplified polynomial. 4. Find x- and y- intercept 5. Determine multiplicity 6. Determine parent function 7. Determine behavior for large values of x

5 Find all seven properties of f (x) = x 2 (x 9). How to find MAX/MIN on Calculator:

6 Section 4.2: Properties of Rational Functions Find the Domain of a Rational Function The denominator of a fraction can NOT equal the value of. Example: Find the domain of the following functions in set builder and interval notation: a) g(x) = 2 x + 3 b) h(x) = 6 x c) f (x) = x 2 +16x + 39

7 Find Asymptotes of Rational Functions I. Vertical Asymptotes Graphs CAN NOT touch or cross a vertical asymptote. If you have the same factor in the numerator and denominator it is NOT a V.A., it is a hole. How to find a V.A.? Set the denominator equal to zero. II. Horizontal Asymptotes

8 If n m = 1, you have an oblique asymptote (slant asymptote). To find: Long or Synthetic Division. Remember you just want the quotient.not the remainder. Example: If a rational function is proper, then is a horizontal asymptote. Example: Find Vertical and Horizontal/Oblique Asymptote if any of R(x) = 6x2 + 5x 6 2x + 3. Example: Find the Vertical and Horizontal/Oblique Asymptote if any of f (x) = x 3 64 x 2 x 12.

9 Transformations of Rational Functions Example:

10 4.3 The Graph of a Rational Function Analyzing the Graph of a Rational Function 1. Factor numerator and denominator. Find the domain of the Rational Function. 2. Locate intercepts of f(x). x-intercepts: set numerator equal to zero y-intercepts: find f(0). 3. Find Asymptotes Vertical asymptotes: set the denominator equal to zero (watch for holes!) Horizontal asymptotes: 3 rules Oblique asymptotes: Long or Synthetic Division 4. Plot points to make graph OR Determine behavior at each vertical asymptote a) substitute VA in every x-value except where it will make the function undefined. b) Simplify c) Match behavior based on 4 graphs Example: f (x) = x +14 x(x +18)

11 Example: g(x) = x2 15x 16 x +19 Example: h(x) = 35x2 18x 81 5x 2 54x + 81

12 Example: A parcel delivery service has contracted you to design a closed box with a square base that has a volume of 10,500 in 3. a) Express the surface area of the box as a function of x and simplify. b) Use a graphing utility to graph s(x). [0, 100, 10] [0, 8000, 1000] c) What is the minimum amount of material that can be used to construct the box? d) What are the dimensions of the box that minimize the surface area?

13 4.4 Polynomial & Rational Inequalities WARM UP: What is the domain of g(x) = x 6 x + 4? Solve Polynomial and Rational Inequalities Steps 1. Get the equation in the form: f (x) < 0; f (x) > 0; f (x) 0; f (x) 0 2. Polynomial: factor in order to determine critical values Rational: combine so there is one fraction. Remember set the numerator and denominator equal to zero for critical values. 3. Make a # line highlighting the critical values. 4. Pick test values in each interval and test those values in the equation from step 2. * f(x) > 0 then * f(x) < 0 then Determine needed interval; Graph solution. * Remember that you do not put [ ] around V.A. s OR infinity symbols. Example: Solve the inequality x 2 4x > 12 Example: Solve the inequality x 4 > 4x 2

14 Example: Solve the following inequality. x 2 (9 + x)(x 7) (x + 4)(x 5) 0 Example: Solve the inequality: 2x x +1 1 Example: Solve the inequality: x +1 x 1 2 Example: Suppose the daily cost C of manufacturing x bicycles is given by C(x) = 40x x Now the average daily cost is given by C(x) =. How many x bicycles must be produced each day for the average cost to be no more then $90?

15 4.5 The Real Zeros of A Polynomial Equation WARM UP: Factor a) 3x 2 + 2x 1 b) 8x 2 + 6x 5 Types of Real Zeros 1. Rational Zeros: Examples: 2. Irrational Zeros: Examples: Use the Remainder and Factor Theorem

16 Example: Determine whether (x 3) is a factor of 3x 4 7x 3 14x 12 Example: Determine whether (x 1) is a factor of 4x 4 2x 3 + 7x 9 Use the Rational Zeros Theorem to list the Potential Rational Zeros of a Polynomial Function Example: Find the Possible Rational Zeros for f (x) = 6x 4 x

17 If the equation does not factor easily To find the Real Zeros of a Polynomial 1. Determine how many zeros the polynomial could have 2. Determine the PRZ s 3. Graph the equation using a graphing utility 4. Extract all zeros that are integers/clearly visible using synthetic until a quadratic or linear equation is left. 5. Find the last zero(s) by factoring, quadratic formula and/or solving linear equations NOTE: Quadratic Formula: x = b ± b2 4ac 2a Example: Find the real zeros of: a) 3x 3 x 2 + 3x 1 b) 3x 4 25x x 2 75x +18 c) 3x 4 58x x 2 483x +117

18 Use the Theorem for Bound on Zeros Example: Find a bound on the real zeros of the polynomial function. a) f (x) = 3x 4 + 6x 3 x 2 12x 15 b) f (x) = 2x 4 + 2x 3 x 2 6x 2 The Intermediate Value Theorem

19 Example: Use the Intermediate Value Theorem to show that the polynomial function has a zero in the given interval. f (x) = 5x 3 + 5x 2 8x + 9; [ 5, 1] Graphing Polynomials using a graphing utility a) f (x) = x 3 + 2x 2 5x 6 [-10, 10, 1] x [-38, 44, 5] b) g(x) = x 4 + x 3 3x 2 x + 2 [-8, 8, 1] x [-42, 42, 100]

20 4.6 Complex Zeros; Fundamental Theorem of Algebra Example: Degree 3; Zeros: 2, -9 i. What is the other zero? Find Polynomials with specified zeros steps 1. Make sure that all zeros are accounted for by referencing degree. 2. List all of the factors. 3. Expand equation using FOIL/Distributive Method 4. Determine Leading Coeffiicient NOTE: i = 1 i 2 = 1 i 3 = i i 4 = 1 Example: Form a polynomial f(x) with real coefficients having the given degree and zeros. a) Degree 4; zeros: 2, multiplicity 2; 6i b) Degree 4; zeros: -2, multiplicity 2; 3 5i

21 Theorem: Every polynomial function of odd degree with real coefficients will have at least one real zero. Find the Complex Zeros of a Polynomial Function Example: Find the complex zeros of the following equations a) f (x) = x 2 18x + 90 b) g(x) = 3x 4 10x 3 12x x 39 (Write answer in factored form). c) h(x) = 5x 4 11x x x 26 (Write answer in factored form).

22 d) f (x) = 5x 5 + 7x x x 2 360x 504 given that -6i is a zero. e) g(x) = 5x 5 + 6x x x 2 135x 162 given that -3i is a zero.