Chapter 3

1. Chapter 3 Polynomial and Rational Functions

2. 3.1 Polynomial Functions and Their Graphs • A polynomial function P is given by where the coefficients are real numbers and the exponents are whole numbers. • The leading term is , the leading coefficient is

3. Behavior of Graphs of Functions • We are interested in the behavior exhibited by the graphs of functions. • We are interested in what the function when x is near the restrictions in the domain. • We are interested what the function does as x becomes very large and very small. • We are interested in the end behavior of the function. The leading term is very influential on the behavior of the graph as x gets very large and when x gets very small.

4. The Leading Term Test n is even andn is even and • If is the leading term of a polynomial, then as or as the graph of the polynomial can be described in one of four ways.

5. The Leading Term Test n is odd and n is odd and

6. Zeros of Polynomials If P is a polynomial and if c is a number such that P(c)=0, then we say c is a zero of P. The following are equivalent ways of saying the same thing. • c is a zero of P. • x = c is a root of the equation P(x)=0. • x-c is a factor of P(x).

7. The Intermediate Value Theorem • For any polynomial function with real coefficients, suppose that for , and have opposite signs. Then there exists some c with such that . • Very important when we are looking for zeros of polynomial functions.

8. Guidelines for Graphing Polynomial Functions • Zeros. Find all zeros, as these are the x-intercepts of the graph. • Test Points. Test values on either side of the zeros to know where the graph is going. • End Behavior. • Sketch the graph.

9. Repeated Linear Factors and Their Effects on the Graph c c c c

10. Important Info from Degree of Polynomial A polynomial of degree n will have … • At most n – 1 local extrema (turning points). • Exactly n linear factors.

11. 3.2 Polynomial Division; The Remainder and Factor Theorems • Our goal is to be able to find all zeros of polynomial functions. • Given any polynomial function, we want to be able to factor the polynomial completely … if it is possible. • Long Division, synthetic division, and two theorems are the focus.

12. The Division Algorithm If P(x) and D(x) are polynomials, with then there exist unique polynomials Q(x) and R(x) such that where R(x) is either 0 or of degree less than the degree of D(x). The polynomials P(x) and D(x) are called the dividend and divisor, respectively, Q(x) is the quotient and R(x) is the remainder.

13. The Remainder Theorem If the polynomial P(x) is divided by x - c, then the remainder is the value P(c)

14. The Factor Theorem c is a zero of P if and only if x – c is a factor of P(x).

15. 3.3 Real Zeros of Polynomials If is a polynomial function with integer coefficients, then if c is a zero of then where p is a factor of the leading coefficient and q is a factor of the constant term. Rational Zeros Theorem

16. Finding the Rational Zeros of a Polynomial • List Possible Zeros. Use the Rational Zeros Theorem. • Divide. Use synthetic division to find the quotient. • Repeat. Repeat steps 1 and 2 for the quotient. Stop when you reach a quadratic. Then factor or use the quadratic formula to find the remaining two zeros.

17. Zeros of Polynomials with Rational Coefficients • If , a and c rational, b not a square, is a zero of a polynomial function with rational coefficients, then is also a zero.

18. Descartes’ Rule of Signs Let P be a polynomial with real coefficients. • The number of positive real zeros of P(x) is either equal to the number of variations in sign in P(x) or is less than that by an even whole number. • The number of negative real zeros is either equal to the number of variations in sign of P(-x) or is less than that by an even whole number.

19. The Upper and Lower Bounds Theorem Let P be a polynomial with real coefficients. • If we divide P(x) by x – b (with b>0), and if the quotient and remainder have no negative entry, then b is an upper bound for the real zeros of P. • If we divide P(x) by x – a (with a<0), and if the quotient and remainder have entries that alternate nonpositive/nonnegative, then a is a lower bound for the zeros of P.

20. 3.4 The Complex Numbers • Consider the solutions to the equation • The solutions are • These are also the x intercepts of the graph.

21. Complex Numbers • What about the equation ? • The solutions are non-real: • What does this mean about the x intercepts of the graph?

22. Definition of “i” and Complex Number • The number i is defined such that and . • A complex number is a number a + bi where a and b are real numbers. • Note: a or b or both can be zero. If b=0, then we have a real number. • If b is not zero, then we have a complex number.

23. The Big Picture of Our Number System Complex Numbers Imaginary Numbers Real Numbers Pure Imaginary Imaginary Numbers Irrational Numbers Rational Numbers

24. Operations on Complex Numbers • Addition and subtraction work like normal. • Multiplication: We must deal with the negative radicand first by “factoring out an i”.

25. Multiplication of Complex Numbers • Remember to FOIL when multiplying two complex numbers together. • The conjugate of a complex number a + bi is a – bi.

26. Division of Complex Numbers • The goal is to remove all i’s from the denominator. • We do this by multiplying the top and bottom by the conjugate of the denominator.

27. 3.5 Complex Zeros and the Fundamental Theorem of Algebra The Fundamental Theorem of Algebra:Every polynomial with complex coefficients has at least one complex zero.

28. Complete Factorization Theorem If P(x) is a polynomial of degree n > 0, then there exist complex numbers such that Zeros Theorem Every polynomial of degree n greater than or equal to 1 has exactly n zeros, provided that a zero of multiplicity k is counted k times.

29. Zeros of Polynomial Functions with Real Coefficients • If a complex number is a zero of a polynomial function with real coefficients, then its conjugate, , is also a zero. Nonreal zeros occur in conjugate pairs. • Freebie: If an irrational number is a zero of a polynomial with integer coefficients, then its conjugate is also a zero.

30. Linear and Quadratic Factors Theorem Every polynomial with real coefficients can be factored into a product of linear and irreducible quadratic factors with real coefficients.

31. 3.6 Rational Functions • A rational function is a function f that is a quotient of two polynomials.Where and are polynomials and where is not the zero polynomial. The domain of f consists of all inputs x for which .

32. Asymptotes • Two types: Vertical Asymptotes and Horizontal Asymptotes. • Vertical Asymptotes come from the restrictions on the domain. The graph will NEVER cross a vertical asymptote. • Horizontal Asymptotes come from the end behavior of the function. A graph MAY cross a horizontal asymptote.

33. Vertical Asymptote • The line is a vertical asymptote for the graph of f if any of the following is true. • Vertical Asymptotes come from the zeros of the denominator.

34. Horizontal Asymptote • The line is a horizontal asymptote for the graph of f if either or both of the following are true:

35. Determining a Horizontal Asymptote • Check and compare the degree of the numerator to the degree of the denominator. • If the deg of numer = deg of denom, the line is the H.A., where a is the leading coefficient of the numerator and b is the leading coefficient of the denominator. • If deg of numer < deg of denom, then is the H.A. • If deg of numer = deg of denom +1, then there is an oblique asymptote and long division must be used.

36. Steps for Graphing a Rational Function • Find domain. • List vertical asymptote(s). • Determine End behavior (horizontal or oblique asymptote) • Find x and y intercepts. • Plot a few more points in between the vertical asymptote(s).