Mathematics 116 Chapter 4 Bittinger

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Mathematics 116 Chapter 4 Bittinger

• Polynomial
• and
• Rational Functions

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Newt Gingrich

• Perseverance is the hard work you do after you
  get tired of doing the hard work you already did.

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Definition of a Polynomial Function

• Polynomial function of x with degree n.

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Joseph De Maistre (1753-1821 French Philosopher

• It is one of mans curious idiosyncrasies to
  create difficulties for the pleasure of resolving
  them.

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Mathematics 116

• Polynomial Functions of Higher Degree

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Continuous

• The graph has no breaks, holes, or gaps.
• Has only smooth rounded turns, not sharp turns
• Its graph can be drawn with pencil without
  lifting the pencil from the paper.

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Leading Coefficient Test

• The leading term determines the end behavior of
  graphs.
• Very Important!

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Objective

• Use the Leading Coefficient Test to determine the
  end behavior of graphs of polynomial functions.

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Intermediate Value Theorem

• Informal Find a value x a at which a
  polynomial function is positive, and anther value
  x b at which it is negative, the function has
  at least one real zero between these two values.
• Use numerical zoom with table or
• Use CAL? 1zero

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Real Zeros of Polynomial Functions

• x a is a zero of function f
• x a is a solution of the polynomial equation
  f(x)0
• (x-a) is a factor of the polynomial f(x)
• (a,0) is an x-intercept of the graph of f.

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Repeated Zeros

• For a polynomial function, a factor
• Yields a repeated zero x a of multiplicity k
• If k is odd, the graph crosses at x a
• If k is even, the graph touches at xa (not cross)

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Objective

• Find and use zeros of polynomial functions as
  sketching aids.

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Chinese Proverb

• A journey of a thousand miles must begin with a
  single step.

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Mathematics 116

• Real Zeros
• of
• Polynomial Functions

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Objective

• Use long division to divide polynomials by other
  polynomials.

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Objective

• Use synthetic division to divide polynomials by
  binomial of the form (x k)

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Reminder Theorem

• If a polynomial f(x) is divided by x k, the
  reminder is r f(k)

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Factor Theorem

• A polynomial f(x) has a factor
• (x k) if and only if f(k) 0

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Using the remainder

• A reminder r obtained by dividing f(x) by x k
• 1. The reminder r gives the value of f at x
  k that is r f(k)
• 2. If r 0, (x k) is a factor of f(x)
• 3. If r 0, the (k,0) is an x intercept of the
  graph of f
• 4. If r 0, then k is a root.

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Rational Roots Test

• Possible rational zeros
• factors of constant term factors of leading
  coefficient
• Possible there are no rational roots.

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Descartes Rule of Signs

• Provides information on number of positive roots
  and number of negative roots.

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William Cullen Bryant (1794-1878) U.S. poet,
editor

• Difficulty, my brethren, is the nurse of
  greatness a harsh nurse, who roughly rocks her
  foster-children into strength and athletic
  proportion.

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Mathematics 116

• The
• Fundamental Theorem
• of
• Algebra

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Number of roots

• A nth degree polynomial has n roots.
• Some of these roots could be multiple roots.

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Linear Factorization Theorem

• Any nth-degree polynomial can be written as the
  product of n linear factors.

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Objective

• Use the fundamental Theorem of Algebra to
  determine the number of zeros (roots) of a
  polynomial function.

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Objective

• Find all zeros of polynomial functions including
  complex zeros.

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Conjugate Roots

• If a bi, where b is not equal to 0 is a
  zero of a function f(x)
• the conjugate a bi is also zero of the
  function.

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John F. Kennedy

• We must use time as a tool, not as a couch.

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Mathematics 116

• Rational Functions
• and
• Asymptotes

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Rational Function
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Graph domain, range, intercepts, asymptotes
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Graph domain, range, intercepts, asymptotes
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Asymptotes

• Vertical
• Horizontal
• Slant

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Objective

• Find the domains of rational functions.

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Objective

• Find horizontal and vertical asymptotes of graphs
  of rational functions.

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Objective

• Use rational functions to model and solve
  real-life problems.

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George S. Patton

• Accept the challenges, so you may feel the
  exhilaration of victory.

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Mathematics 116

• Graphs of a Rational Function

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Graphing Rational Function

• 1. Simplify f if possible reduce
• 2. Evaluate f(0) for y intercept and plot
• 3. Find zeros or x intercepts set numerator
  0 solve
• 4. Find vertical asymptotes set denominator
  0 and solve
• 5. Find horizontal / slant asymptotes
• 6. Find holes

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Dan Rather

• Courage is being afraid but going on anyhow.

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College Algebra 116

• Quadratic Inequalities

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Sample Problem quadratic inequalities 1
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Sample Problem quadric inequalities 2
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Sample Problem quadratic inequalities 3
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Sample Problem quadratic inequalities 4
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Sample Problem quadratic inequalities 5
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Everette Dennis Media professor

• Theres a compelling reason to master
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