Download Section 4.2 Complex Solutions of Equations

82
Chapter 4
Complex Numbers
Course Number
Section 4.2 Complex Solutions of Equations
Instructor
Objective: In this lesson you learned how to determine the number of
zeros of polynomial functions, and to find the zeros.
I. The Number of Solutions of a Polynomial Equation
(Pages 335−336)
The Fundamental Theorem of Algebra implies that a polynomial
equation of degree n has
precisely n
Date
What you should learn
How to determine the
numbers of solutions of
polynomial equations
solutions in the
complex number system. These solutions can be . . .
real or
complex and may be repeated.
Example 1: How many zeros does the polynomial function
f ( x) = 5 − 2 x 2 + x 3 − 12 x 5 have?
5
You can use a graph to check the number of
real
solutions of an equation.
Every second-degree equation, ax 2 + bx + c = 0 , has precisely
two solutions given by the Quadratic Formula. The expression
b2 − 4ac
is called the discriminant, and can be used
to determine whether the solutions are real, repeated, or
complex:
1) If the discriminant is less than zero, the equation has
two complex
solution(s).
2) If the discriminant is equal to zero, the equation has
one repeated real
solution(s).
3) If the discriminant is greater than zero, the equation has
two distinct real
solution(s).
Larson/Hostetler Trigonometry, Sixth Edition Student Success Organizer IAE
Copyright © Houghton Mifflin Company. All rights reserved.
Section 4.2
83
Complex Solutions of Equations
Example 2: Use the discriminant to find the number and type
of solutions of the quadratic equation
x 2 − 2x + 2 = 0 .
Two complex solutions
II. Finding Solutions of Polynomial Equations (Page 337)
If the complex number a + bi (where b ≠ 0) is a solution of a
What you should learn
How to find the solutions
of polynomial equations
polynomial equation with real coefficients, then we know that
a − bi
is another solution of the equation.
Example 3: Find the solutions of the quadratic equation
x 2 − 2x + 2 = 0 .
The solutions are 1 ± i.
III. Finding Zeros of Polynomial Functions (Pages 338−339)
The problem of finding the zeros of a polynomial function is
essentially the same as . . .
finding the solutions of a
polynomial equation.
The zeros of the polynomial function f ( x) = x 5 − 3 x 2 + 4 are
simply . . .
the solutions of the polynomial equation
x5 − 3x2 + 4 = 0.
Example 4: Find all the zeros of the polynomial function
f ( x) = x 4 + 5 x 2 − 36 , given that 3i is a zero of f.
Zeros: − 2, 2, − 3i, 3i
Example 5: Find a fourth-degree polynomial function with real
coefficients that has − 5, 5, and − 2i as zeros.
f (x) = x4 − 21x2 − 100
Larson/Hostetler Trigonometry, Sixth Edition Student Success Organizer IAE
Copyright © Houghton Mifflin Company. All rights reserved.
What you should learn
How to find the zeros of
polynomial functions
84
Chapter 4
Complex Numbers
Additional notes
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Homework Assignment
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Larson/Hostetler Trigonometry, Sixth Edition Student Success Organizer IAE
Copyright © Houghton Mifflin Company. All rights reserved.