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Complex Numbers Lesson 3.3

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**It's any number you can imagine**

The Imaginary Number i By definition Consider powers if i It's any number you can imagine

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Using i Now we can handle quantities that occasionally show up in mathematical solutions What about

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**Complex Numbers Combine real numbers with imaginary numbers Examples**

a + bi Examples Real part Imaginary part

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Try It Out Write these complex numbers in standard form a + bi

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**Operations on Complex Numbers**

Complex numbers can be combined with addition subtraction multiplication division Consider

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**Operations on Complex Numbers**

Division technique Multiply numerator and denominator by the conjugate of the denominator

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**Complex Numbers on the Calculator**

Possible result Reset mode Complex format to Rectangular Now calculator does desired result

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**Complex Numbers on the Calculator**

Operations with complex on calculator Make sure to use the correct character for i. Use 2nd-i

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**Warning Consider It is tempting to combine them**

The multiplicative property of radicals only works for positive values under the radical sign Instead use imaginary numbers

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Try It Out Use the correct principles to simplify the following:

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**The Discriminant Return of the**

Consider the expression under the radical in the quadratic formula This is known as the discriminant What happens when it is Positive and a perfect square? Positive and not a perfect square? Zero Negative ? Complex roots

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**Example Consider the solution to Note the graph**

No intersections with x-axis Using the solve and csolve functions

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**Fundamental Theorem of Algebra**

A polynomial f(x) of degree n ≥ 1 has at least one complex zero Remember that complex includes reals Number of Zeros theorem A polynomial of degree n has at most n distinct zeros Explain how theorems apply to these graphs

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**Conjugate Zeroes Theorem**

Given a polynomial with real coefficients If a + bi is a zero, then a – bi will also be a zero

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Assignment Lesson 3.3 Page 211 Exercises 1 – 78 EOO

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