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**Sec 3.4 & Sec 3.5 Complex Numbers & Complex Zeros**

Objectives: To understand complex numbers. To add, subtract & multiply complex numbers. To solve equations with complex numbers as solutions.

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Complex Numbers If , then A complex number is a number of the form where a and b are real numbers. The real part of the complex number is a and the imaginary part is bi.

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**Ex 1. State the real and imaginary parts of the following complex numbers.**

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**Operations on Complex Numbers**

To add complex numbers, add the real parts and add the imaginary parts. (a + bi) + (c + di) = (a + c) + (b + d)I To subtract complex numbers, subtract the real parts and subtract the imaginary parts. (a + bi) – (c + di) = (a – c) + (b – d)i

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**Multiply complex numbers like binomials, using**

(a + bi) . (c + di) = (ac – bd) + (ad + bc)i

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**Ex 2. Perform the indicated operation and write the result as a + bi.**

(3 + 5i) + (4 – 2i) (3 + 5i) – (4 – 2i) (3 + 5i)(4 – 2i)

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**Powers of Any power of i can be reduced to one of the following:**

To reduce a power of i, just divide by 4 and the remainder will correspond to one of these. For example, Because 23/4 has a remainder of 3 so the answer corresponds to

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Ex 3. Evaluate. a) b) c) d)

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Class Work Perform the indicated operation and write your answer in a + bi form. (2 + 5i) + (4 – 6i) (5 – 3i)(1 + i)

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**Square Roots of Negative Numbers**

Just as every positive real number r has two square roots , every negative number has two square roots as well. If -r is a negative number, then its square roots are , because: and

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**If –r is negative, then the principal square root of –r is: For example: **

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**Ex 4. Evaluate the expression.**

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**Ex 5. Evaluate and write the result in the form of a + bi.**

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Class Work Evaluate the expression and write your result in the form of a + bi

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**Complex Roots of Quadratic Equations**

We know that if b2 – 4ac < 0 then the quadratic has no real solutions. However, in the complex number system, the equation will always have solutions. The solutions will be complex numbers and will have the form a + bi and a – bi. The solutions will always come in pairs, called conjugates.

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Ex 6. Solve each equation. x2 + 9 = 0 x2 + 4x + 5 = 0

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Class Work Find all the zeros of the polynomial

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**Ex 7. Find a polynomial with integer coefficients that have the given zeros.**

Degree 3 and zeros 2 and i. Degree 2 and zeros 1 + i, and 1 – i.

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HW Worksheet 3.4 & 3.5

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