2.7 Apply the Fundamental Theorem of Algebra

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2.7 Apply the Fundamental Theorem of Algebra
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• What were doing today
• Learning what the Fundamental Theorem of Algebra
  says about polynomials.
• Learn how to apply the fundamental Theorem of
  Algebra with real, imaginary, and complex zeros.

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The Fundamental Theorem of Algebra If f(x) is a
polynomial and f(x) has degree n (ngt0),then f(x)
0 has at least one solution in the complex
plane. Corollary f(x) with degree n (ngt0) has
exactly n solutions.
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Example 1
a. How many solutions does the equation
x4 5x2 36 0 have?
b. How many zeros does the function f
(x) x3 7x2 8x 16 have?
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Find all the zeros of the polynomial.
Example 2
f (x) x5 2x4 8x2 13x 6
f (x) (x 1)2 (x2) (x2 2x 3)
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Example 2
f (x) x5 2x4 8x2 13x 6
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Find a polynomial function with the given zeros.
Example 3
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Example 4
Use the zero command on your graphing calculator
to approximate the real zeros of the function.
 
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1. What is the degree of f (x) 8x6 4x5 3x2
2?
ANSWER
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2. Solve x2 2x 3 0
ANSWER
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3. The function P given by x4 3x3 30x2 6x
56 model the profit of a company. What are the
real solution of the function?
ANSWER
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EXAMPLE 1
Find the number of solutions or zeros
a. How many solutions does the equation
x3 5x2 4x 20 0 have?
SOLUTION
Because x3 5x2 4x 20 0 is a polynomial
equation of degree 3,it has three solutions.
(The solutions are 5, 2i, and 2i.)
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EXAMPLE 1
Find the number of solutions or zeros
b. How many zeros does the function f
(x) x4 8x3 18x2 27 have?
SOLUTION
Because f (x) x4 8x3 18x2 27 is a
polynomial function of degree 4, it has four
zeros. (The zeros are 1, 3, 3, and 3.)
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EXAMPLE 2
Find the zeros of a polynomial function
Find all zeros of f (x) x5 4x4 4x3 10x2
13x 14.
SOLUTION
f (x) (x 1)2(x 2)(x2 4x 7)
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EXAMPLE 2
Find the zeros of a polynomial function
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for Example 2
GUIDED PRACTICE
Find all zeros of the polynomial function.
3. f (x) x3 7x2 15x 9
SOLUTION
Formula are (x 1)2 (x 3)
f(x) (x 1) (x 3)2
The zeros of f are 1 and 3
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EXAMPLE 3
Use zeros to write a polynomial function
SOLUTION
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EXAMPLE 3
Use zeros to write a polynomial function
Write f (x) in factored form.
Regroup terms.
(x 3)(x 2)2 5
Multiply.
(x 3)(x2 4x 4) 5
Expand binomial.
(x 3)(x2 4x 1)
Simplify.
x3 4x2 x 3x2 12x 3
Multiply.
x3 7x2 11x 3
Combine like terms.
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GUIDED PRACTICE
for Example 3
Write a polynomial function f of least degree
that has rational coefficients, a leading
coefficient of 1, and the given zeros.
5. 1, 2, 4
Use the three zeros and the factor theorem to
write f(x) as a product of three factors.
SOLUTION
f (x) (x 1) (x 2) ( x 4)
Write f (x) in factored form.
(x 1) (x2 4x 2x 8)
Multiply.
(x 1) (x2 6x 8)
Combine like terms.
x3 6x2 8x x2 6x 8
Multiply.
x3 5x2 2x 8
Combine like terms.
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GUIDED PRACTICE
for Example 3
SOLUTION
Write f (x) in factored form.
Regroup terms.
Multiply.
(x 2)(x2 4)(x2 8x16 6)
Expand binomial.
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GUIDED PRACTICE
for Example 3
(x 2)(x2 4)(x2 8x 10)
Simplify.
(x2) (x4 8x2 10x2 4x2 3x 40)
Multiply.
(x2) (x4 8x3 14x2 32x 40)
Combine like terms.
x5 8x4 14x3 32x2 40x 2x4 16x3 28x2
64x 80
Multiply.
x510x4 30x3 60x2 10x 80
Combine like terms.
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GUIDED PRACTICE
for Example 3
8. 3, 3 i
Because the coefficients are rational and 3 i is
a zero, 3 i must also be a zero by the complex
conjugates theorem. Use the three zeros and the
factor theorem to write f(x) as a product of
three factors
SOLUTION
f(x) (x 3)x (3 i)x (3 i)
Write f (x) in factored form.
(x3)(x 3)i (x2 3) i
Regroup terms.
(x3)(x 3)2 i2)
Multiply.
(x 3)(x 3) i(x 3) i
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GUIDED PRACTICE
for Example 3
(x 3)(x 3)2 i2(x 3)(x2 6x 9)
Simplify.
(x3)(x2 6x 9)
x36x2 9x 3x2 18x 27
Multiply.
Combine like terms.
x3 9x2 27x 27
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EXAMPLE 4
Use Descartes rule of signs
Determine the possible numbers of positive real
zeros, negative real zeros, and imaginary zeros
for f (x) x6 2x5 3x4 10x3
6x2 8x 8.
SOLUTION
f (x) x6 2x5 3x4 10x3 6x2 8x 8.
The coefficients in f (x) have 3 sign changes, so
f has 3 or 1 positive real zero(s).
f ( x) ( x)6 2( x)5 3( x)4 10( x)3
6( x)2 8( x) 8
x6 2x5 3x4 10x3 6x2 8x 8
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EXAMPLE 4
Use Descartes rule of signs
The coefficients in f ( x) have 3 sign changes,
so f has 3 or 1 negative real zero(s) .
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for Example 4
GUIDED PRACTICE
Determine the possible numbers of positive real
zeros, negative real zeros, and imaginary zeros
for the function.
9. f (x) x3 2x 11
SOLUTION
f (x) x3 2x 11
The coefficients in f (x) have 1 sign changes, so
f has 1 positive real zero(s).
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for Example 4
GUIDED PRACTICE
f ( x) ( x)3 2( x) 11
x3 2x 11
The coefficients in f ( x) have no sign changes.
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for Example 4
GUIDED PRACTICE
10. g(x) 2x4 8x3 6x2 3x 1
SOLUTION
f (x) 2x4 8x3 6x2 3x 1
The coefficients in f (x) have 4 sign changes, so
f has 4 positive real zero(s).
f ( x) 2( x)4 8( x)3 6( x)2 1
2x4 8x 6x2 1
The coefficients in f ( x) have no sign changes.
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for Example 4
GUIDED PRACTICE
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EXAMPLE 5
Approximate real zeros
Approximate the real zeros of f (x) x6 2x5
3x4 10x3 6x2 8x 8.
SOLUTION
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EXAMPLE 6
Approximate real zeros of a polynomial model
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EXAMPLE 6
Approximate real zeros of a polynomial model
SOLUTION
Substitute 15 for s(x) in the given function. You
can rewrite the resulting equation as
0 0.00547x3 0.225x2 3.62x 26.0
Then, use a graphing calculator to approximate
the real zeros of f (x) 0.00547x3 0.225x2
3.62x 26.0.
From the graph, there is one real zero x 19.9.
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for Examples 5 and 6
GUIDED PRACTICE

Approximate the real zeros of f (x) 3x5 2x4
8x3 4x2 x 1.
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for Examples 5 and 6
GUIDED PRACTICE
12. What If? In Example 6, what is the
tachometer reading when the boat
travels 20 miles per hour?
SOLUTION
Substitute 20 for s(x) in the given function. You
can rewrite the resulting equation as
0 0.00547x3 0.225x2 3.62x 31.0
Then, use a graphing calculator to approximate
the real zeros of f (x) 0.00547x3 0.225x2
3.62x 31.0.
From the graph, there is one real zero x 23.1.
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for Examples 5 and 6
GUIDED PRACTICE
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Daily Homework Quiz

1. Find all the zeros of f(x) x4 x2 20.
2. Write a polynomial function of least degree
that has rational coefficients, a leading
coefficient of 1, and 3 and 1 7i
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Daily Homework Quiz

3. Determine the possible numbers of positive
real zeros, negative real zeros, and imaginary
zeros for f(x) 2x5 3x4 5x3 10x2
3x 5.
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Daily Homework Quiz

4. The profit P for printing envelopes is
modeled by P x 0.001x3 0.06x2
30.5x, where x is the number of envelopes
printed in thousands. What is the least
number of envelopes that can be printed for
a profit of 1500?