6.7 Using the Fundamental Theorem of Algebra

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6.7 Using the Fundamental Theorem of Algebra
What is the fundamental theorem of Algebra? What
methods do you use to find the zeros of a
polynomial function? How do you use zeros to
write a polynomial function?
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• German mathematician Carl Friedrich Gauss
  (1777-1855) first proved this theorem. It is the
  Fundamental Theorem of Algebra.

If f(x) is a polynomial of degree n where n gt 0,
then the equation f(x) 0 has at least one root
in the set of complex numbers.
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Solve each polynomial equation. State how many
solutions the equation has and classify each as
rational, irrational or imaginary.
x ½, 1 sol, rational

• 2x -1 0
• x2 -2 0
• x3 - 1 0

(x -1)(x2 x 1), x 1 and use Quadratic
formula for
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Solve the Polynomial Equation.

• x3 x2 -x - 1 0

Notice that -1 is a solution two times. This is
called a repeated solution, repeated zero, or a
double root.

• 1 -1 -1

1
1
2
1
1
2
1
0
x2 2x 1 (x 1)(x 1) x -1, x -1, x 1
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Finding the Number of Solutions or Zeros
x
3

• x3 3x2 16x 48 0
• (x 3)(x2 16) 0
• x 3 0, x2 16 0
• x -3, x2 -16
• x - 3, x 4i

x3
3x2
x2
16x
48
16
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Finding the Number of Solutions or Zeros

• f(x) x4 6x3 12x2 8x
• f(x) x(x3 6x2 12x 8)
• 8/1 8/1, 4/1, 2/1, 1/1
• Synthetic division
• x3 6x2 12x 8
• 1 6 12 8

Zeros -2,-2,-2, 0
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Finding the Zeros of a Polynomial Function

• Find all the zeros of f(x) x5 - 2x4 8x2 - 13x
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Possible rational zeros 6, 3, 2, 1 1 -2
0 8 -13 6
1
1
-1
7
-6
-1
1
-1
-1
7
0
-6
-2
-2
6
-10
6
1
-3
5
-3
0
1
1
-2
3
1
-2
3
0
x2 -2x 3 Use quadratic formula
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Graph of polynomial function

• Turn to page 367 in your book.
• Real zero where the graph crosses the x-axis.
• Repeated zero where graph touches x-axis.

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Using Zeros to Write Polynomial Functions

• Write a polynomial function f of least degree
  that has real coefficients, a leading coefficient
  of 1, and 2 and 1 i as zeros.
• x 2, x 1 i, AND x 1 - i.
• Complex conjugates always travel in pairs.
• f(x) (x - 2)x - (1 i )x - (1 - i )
• f(x) (x - 2)(x - 1) - i (x - 1) i
• f(x) (x - 2)(x - 1)2 - i2
• f(x) (x - 2)(x2 - 2x 1 -(-1)
• f(x) (x - 2)x2 - 2x 2
• f(x) x3 - 2x2 2x - 2x2 4x - 4
• f(x) x3 - 4x2 6x - 4

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• What is the fundamental theorem of Algebra?
• If f(x) is a polynomial of degree n where n gt 0,
  then the equation f(x) 0 has at least one root
  in the set of complex numbers.
• What methods do you use to find the zeros of a
  polynomial function?
• Rational zero theorem (6.6) and synthetic
  division.
• How do you use zeros to write a polynomial
  function?
• If x , it becomes a factor (x ). Multiply
  factors together to find the equation.

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Assignment is p. 369, 15-29 odd, 35-43 odd Show
your work