LESSON 13 - POLYNOMIAL EQUATIONS & COMPLEX NUMBERSPreCalculus - Santowski

LESSON OBJECTIVES

Review the basic idea behind complex number system

Review basic operations with complex numbers

Find and classify all real and complex roots of a polynomial equation

Write equations given information about the roots

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FAST FIVE

STORY TIME.....

http://mathforum.org/johnandbetty/frame.htm

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(A) INTRODUCTION TO COMPLEX NUMBERS

Solve the equation x2 – 1 = 0

We can solve this many ways (factoring, quadratic formula, completing the square & graphically)

In all methods, we come up with the solution of x = + 1, meaning that the graph of the quadratic (the parabola) has 2 roots at x = + 1.

Now solve the equation x2 + 1= 0

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(A) INTRODUCTION TO COMPLEX NUMBERS

the equation x2 + 1= 0 has no roots because you cannot take the square root of a negative number.

Long ago mathematicians decided that this was too restrictive They did not like the idea of an equation having no solutions -- so they invented them.

They proved to be very useful, even in practical subjects like engineering.

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(A) INTRODUCTION TO COMPLEX NUMBERS

since solutions are “invented” then let’s “invent” a symbol to help us

We shall use the symbol i to help us out and define i as …

So then, if or

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1or 12 ii

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(A) INTRODUCTION TO COMPLEX NUMBERS

Note the difference (in terms of the expected solutions) between the following 2 questions:

Solve x2 + 5 = 0 where Solve x2 + 5 = 0 where

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Rx

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(B) OPERATIONS WITH COMPLEX NUMBERS

So if we are going to “invent” a number system to help us with our equation solving, what are some of the properties of these “complex” numbers?

How do we operate (add, sub, multiply, divide)

How do we graphically “visualize” them? Powers of i Absolute value of complex numbers

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(B) OPERATIONS WITH COMPLEX NUMBERS

Exercise 1. Add or subtract the following complex numbers using your TI-84 (switch to complex mode).

(a) (3 + 2i) + (3 + i) (b) (4 − 2i) − (3 − 2i) (c) (−1 + 3i) + (2 + 2i) (d) (2 − 5i) − (8 − 2i)

Generalize HOW do you add complex numbers?

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(B) OPERATIONS WITH COMPLEX NUMBERS

to add or subtract two complex numbers, z1 = a + ib and z2 = c+id, the rule is to add the real and imaginary parts separately:

z1 + z2 = a + ib + c + id = a + c + i(b + d) z1 − z2 = a + ib − c − id = a − c + i(b − d)

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(B) OPERATIONS WITH COMPLEX NUMBERS Exercise 2. Multiply the following complex

numbers using your TI-84 (switch to complex mode).

(a) (3 + 2i)(3 + i) (b) (4 − 2i)(3 − 2i) (c) (−1 + 3i)(2 + 2i) (d) (2 − 5i)(8 − 3i) (e) (2 − i)(3 + 4i)

Generalize HOW do you multiply complex numbers?

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(B) OPERATIONS WITH COMPLEX NUMBERS We multiply two complex numbers just as we

would multiply expressions of the form (x + y) together

(a + ib)(c + id) = ac + a(id) + (ib)c + (ib)(id) = ac + iad + ibc − bd = ac − bd + i(ad + bc)

Example (2 + 3i)(3 + 2i) = 2 × 3 + 2 × 2i + 3i × 3 + 3i × 2i = 6 + 4i + 9i − 6 = 13i

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(C) COMPLEX CONJUGATION For any complex number, z = a+ib, we define

the complex conjugate to be: . It is very useful since the following are real: = a + ib + (a − ib) = 2a = (a + ib)(a − ib) = a2 + iab − iab − (ib)2 =

a2 + b2

Exercise 4. Combine the following complex numbers and their conjugates and find and

(a) If z = (3 + 2i) (b) If z = (3 − 2i) (c) If z = (−1 + 3i) (d) If z = (4 − 3i)

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ibaz

zz zz

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(D) SOLVING EQUATIONS USING COMPLEX NUMBERS

Note the difference (in terms of the expected solutions) between the following 2 questions:

Solve x2 + 2x + 5 = 0 where Solve x2 + 2x + 5 = 0 where

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Rx

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(D) SOLVING EQUATIONS USING COMPLEX NUMBERS

Solve the following quadratic equations where

Simplify all solutions as much as possible Rewrite the quadratic in factored form

x2 – 2x = -10 3x2 + 3 = 2x 5x = 3x2 + 8 x2 – 4x + 29 = 0

Now verify your solutions algebraically!!!

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(D) SOLVING EQUATIONS USING COMPLEX NUMBERS

One root of a quadratic equation is 3i

(a) What is the other root? (b) What are the factors of the quadratic? (c) If the y-intercept of the quadratic is 6,

determine the equation in factored form and in standard form.

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(D) SOLVING EQUATIONS USING COMPLEX NUMBERS

One root of a quadratic equation is 2 + 3i

(a) What is the other root? (b) What are the factors of the quadratic? (c) If the y-intercept of the quadratic is 6,

determine the equation in factored form and in standard form.

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(D) SOLVING POLYNOMIAL EQUATIONS USING COMPLEX NUMBERS

Solve the following polynomial equations where

Simplify all solutions as much as possible Rewrite the polynomial in factored form

x3 – 2x2 + 9x = 18 x3 + x2 = – 4 – 4x 4x + x3 = 2 + 3x2

Now verify your solutions algebraically!!!

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(D) SOLVING POLYNOMIAL EQUATIONS USING COMPLEX NUMBERS

For the following polynomial functions State the other complex root Rewrite the polynomial in factored form Expand and write in standard form

(i) one root is -2i as well as -3 (ii) one root is 1 – 2i as well as -1 (iii) one root is –i, another is 1-i

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