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Chapter 3. Polynomial and Rational Functions. 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 . Behavior of Graphs of Functions.

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slide1

Chapter 3

Polynomial and Rational Functions

slide2

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
slide3

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.
slide4

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.
slide5

The Leading Term Test

n is odd and

n is odd and

zeros of polynomials
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).
slide7

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.
guidelines for graphing polynomial functions
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.
important info from degree of polynomial
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.
3 2 polynomial division the remainder and factor theorems
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.
the division algorithm
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.

the remainder theorem
The Remainder Theorem

If the polynomial P(x) is divided by x - c, then the remainder is the value P(c)

the factor theorem
The Factor Theorem

c is a zero of P if and only if x – c is a factor of P(x).

3 3 real zeros of polynomials
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

finding the rational zeros of a polynomial
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.
zeros of polynomials with rational coefficients
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.
descartes rule of signs
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.
the upper and lower bounds theorem
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.
3 4 the complex numbers
3.4 The Complex Numbers
  • Consider the solutions to the equation
  • The solutions are
  • These are also the x intercepts of the graph.
complex numbers
Complex Numbers
  • What about the equation ?
  • The solutions are non-real:
  • What does this mean about the x intercepts of the graph?
definition of i and complex number
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.
the big picture of our number system
The Big Picture of Our Number System

Complex Numbers

Imaginary Numbers

Real Numbers

Pure Imaginary

Imaginary Numbers

Irrational Numbers

Rational Numbers

operations on complex numbers
Operations on Complex Numbers
  • Addition and subtraction work like normal.
  • Multiplication: We must deal with the negative radicand first by “factoring out an i”.
multiplication of complex numbers
Multiplication of Complex Numbers
  • Remember to FOIL when multiplying two complex numbers together.
  • The conjugate of a complex number a + bi is a – bi.
division of complex numbers
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.
3 5 complex zeros and the fundamental theorem of algebra
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.

complete factorization theorem
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.

zeros of polynomial functions with real coefficients
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.

linear and quadratic factors theorem
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.

3 6 rational functions
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 .
asymptotes
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.
vertical asymptote
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.
horizontal asymptote
Horizontal Asymptote
  • The line is a horizontal asymptote for the graph of f if either or both of the following are true:
determining a horizontal asymptote
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.
steps for graphing a rational function
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).