4-1 Polynomial Functions

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4-1Polynomial Functions
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Objectives

• Determine roots of polynomial equations.
• Apply the Fundamental Theorem of Algebra.

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Polynomial in One Variable
a0, a1, a2, . . . an complex numbers (real or
imaginary) a0?0 n non-negative integer
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Definitions

• The degree of a polynomial is the greatest
  exponent of the variable.
• The leading coefficient is the coefficient of the
  variable with the greatest exponent.
• If all the coefficients are real numbers, it is a
  polynomial function.
• The values of x where f(x)0 are called the zeros
  (x-intercepts).

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Example
What is the degree? 4 What is the leading
coefficient? 3 Is -2 a zero of f(x)? no
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Imaginary Numbers
Complex numbers abi (a and b are real
numbers) Pure imaginary number a0 and b?0
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Imaginary Numbers
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Fundamental Theorem of Algebra

• Every polynomial equation with degreegt0 has at
  least one root in the set of complex numbers
• Corollary A polynomial of degree n has exactly n
  complex roots.

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Maximum Number of Roots
Degree 1
Degree 2
Degree 3
Degree 4
Degree 5
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Determining Roots

• You cant determine imaginary roots from the
  graph you can only see the real roots.
• Imaginary roots come in pairs.
• An equation with odd degree must have a real root.

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Finding the Polynomial

• If you know the roots, you can find the
  polynomial.
• (x-a)(x-b)0

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Example

• Write a polynomial equation of least degree with
  roots 2, 3i and -3i.
• (x-2)(x-3i)(x3i)0
• (x-2)(x2-9i 2)0
• (x-2)(x29)0
• x3-2x29x-180
• Does the equation have odd or even degree?
• Odd
• How many times does the graph cross the x-axis?
• Once

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Example

• State the number of complex roots of the equation
  32x3 - 32x2 4x - 40. Find the roots and graph.
• 32x3 - 32x2 4x - 40
• 32x2(x-1)4(x-1)0
• (32x24)(x-1)0

x2-4/32 or x1
xv-1/8
xi/2v2 iv2/4
Roots are 1 and iv2/4
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Meterorology Example (5)

• A meteorologist sends a temperature probe on a
  small weather rocket through a cloud layer. The
  launch pad for the rocket is 2 feet off the
  ground. The height of the rocket after launching
  is modeled by the equation h-16t2232t2
  where h is the height of the rocket in feet and t
  is the elapsed time in seconds. When will the
  rocket be 114 feet above the ground?

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Solution

• Find t when h114.
• h-16t2232t2
• 114-16t2232t2
• 0-16t2232t-112
• 0-8(2t2-29t14)
• 0-8(2t-1)(t-14)
• 2t-10 or t-140
• t1/2 or t14

(xscl5, yscl100)
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Homework

• page 210
• 15-47 odds
• Dont graph 39-47