Polynomial and Rational Functions

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Chapter 3
Polynomial and Rational Functions
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3.1
Quadratic Functions and Models
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Polynomial Function

• A polynomial function of degree n, where n is a
  nonnegative integer, is a function defined by an
  expression of the form
• where an, an ? 1, a1, and a0 are real numbers,
  with an ? 0.
• Examples that are polynomials
• Examples that are NOT polynomials

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Quadratic Functions

• A function f is a quadratic function if
• f(x) ax2 bx c, where a, b, and c are real
  numbers with a ? 0.
• The simplest quadratic function is f(x) x2.
  This is the graph of a parabola.
• The line of symmetry for a parabola is called the
  axis of the parabola.
• The vertex is the point where the axis intersects
  the parabola.

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Example

• Graph the function and find the domain, range
  vertex

• a)

D R V
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Example

• Graph the function and find the domain, range
  vertex

• a)

D R V
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Example

• Graph the function and find the domain, range
  vertex

• a)

D R V
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Completing the Square

• Graph f(x) x2 ? 4x 7 by completing the square
  and locating the vertex.

0 x2 4x 7
-7 x2 4x
-7 4 x2 4x 4
-3 (x 2)2
0 (x 2)2 3
f(x) (x 2)2 3
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Completing the Square continued
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Example

• Graph f(x) ?3x2 6x ? 1 by completing the
  square and locating the vertex.

0 -3x2 6x 1
0 x2 2x 1/3
-1/3 x2 2x
-1/3 1 x2 2x 1
2/3 (x 1)2
0 (x 1)2 2/3
f(x) -3(x 1)2 2
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Example continued

• The vertex is (1, 2). Find additional points by
  substituting x-values into the original equation.

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Graph of a Quadratic Function

• The quadratic function defined by f(x) ax2
  bx c can be written in the form
• f(x) a(x ? h)2 k, a ? 0
• The graph of f has the following characteristics.
• 1. It is a parabola with vertex (h, k). The
• x-coordinate of the vertex can be found
  by

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Graph of a Quadratic Function continued

• 2. It opens up if a gt 0 and down if a lt 0.
• 3. It is the graph of y x2 shrunk
• vertically if a lt 1 and stretched
• vertically if a gt 1.
• 4. The y-intercept is f(0) c.
• 5. If b2 ? 4ac ? 0, the x-intercepts are
• If b2 ? 4ac lt 0, there are no x-intercepts.

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Example

• Find the vertex of the parabola having equation
  f(x) 4x2 8x 3

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Quadratic Inequalities

• For what x-values is ?

-10, 10, 1 by -10, 10, 1
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Quadratic Inequalities

• For what x-values is ?

-10, 10, 1 by -10, 10, 1
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Quadratic Inequalities

• For what x-values is ?

-5, 5, 1 by -10, 10, 1
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HOMEWORK

• Pg 303
• 1 57 eoo

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3.2
Synthetic Division
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Synthetic Division

• A shortcut method of performing long division
  with certain polynomials, called synthetic
  division, is used only when a polynomial is
  divided by a first-degree binomial of the form
  x ? k, where the coefficient of x is 1.

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Caution

• Note To avoid errors, use 0 as coefficient for
  any missing terms, including a missing constant,
  when setting up the division.

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Example
Long Division
Synthetic Division
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Example continued

• We have shown that

Since f(x) 2x3 3x2 32, the remainder, -28,
is the function value of -2, f(-2) -28. This
property is known as the Remainder Theorem.
The Remainder Theorem also says if the remainder
is zero, then k is considered to be a zero of
the given function.
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Example

• Use synthetic division to divide
• 2x3 ? 3x2 ? 11x 7
  by x ? 3.
• Solution Since x ? 3 in the form x ? k use this
  and the coefficients of the polynomial to obtain

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HOMEWORK

• Pg 319
• 1 45 eoo