l esson 13 - p olynomial e quations & c omplex n umbers precalculus - santowski
TRANSCRIPT
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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(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
Cx
(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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