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Objectives: 1. Write complex numbers in standard form. 2. Perform arithmetic operations on complex numbers. 3. Find the conjugate of a complex number. 4. Simplify square roots of negative numbers. 5. Find all solutions of polynomial equations.

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Imaginary & Complex Numbers The imaginary unit is defined as Imaginary numbers can be written in the form bi where b is a real number. A complex number is a sum of a real and imaginary number written in the form a + bi. Any real number can be written as a complex number: Example: 2 = 2 + 0i, 3 = 3 + 0i

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Example #1 Equating Two Complex Numbers Find x and y. To solve make two equations equating the real parts and imaginary parts separately.

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Example #2 Adding, Subtracting, & Multiplying Complex Numbers Perform the indicated operation and write the result in the form a + bi. A. B. Combine like terms. Distribute the (-) and then combine like terms.

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Example #2 Adding, Subtracting, & Multiplying Complex Numbers Perform the indicated operation and write the result in the form a + bi. C. D. Distribute & Simplify. Remember: Use FOIL, substitute and combine like terms.

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Example #3 Products & Powers of Complex Numbers Perform the indicated operation and write the result in the form a + bi. A. B. Since these groups are the same but with opposite signs they are conjugates of each other. The middle terms always cancel with conjugates.

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Powers of i This pattern of {i, 1, i, 1} will continue for even higher patterns. A shortcut to evaluating higher powers requires you to memorize this pattern, but it is not necessary to evaluate them.

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Example #4 Powers of i Find the following: A. Method 1: 1.If the exponent is odd, first break off an i from the original term. 2.Rewrite the even exponent as a power of i 2 (divide it by 2). 3.Replace the i 2 with 1. 4.Evaluate the power on 1. Even exponents make it positive and odd exponents keep it negative. 5.Multiply what is left back together.

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Example #4 Powers of i Find the following: A. Method 2: 1.Use long division and divide the exponent by 4. Always use 4 which is the four possible values of the pattern {i, 1, i, 1}. 2.The remainder represents the term number in the sequence. For this problem the remainder is 1 which means the answer is the first term in the sequence {i, 1, i, 1} which is i.

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Example #4 Powers of i Find the following: B. This time it isnt necessary to break off any i because the exponent is already even. If the remainder is 0 this indicates that the value is the 4 th term in the sequence {i, 1, i, 1} since you cant have a remainder of 4 when dividing by 4. Therefore, the answer is 1.

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Example #4 Powers of i Find the following: C. This time it is necessary to break off an i because the exponent is odd. If the remainder is 3 this indicates that the value is the 3 rd term in the sequence {i, 1, i, 1}. Therefore, the answer isi.

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Example #4 Powers of i Find the following: D. If the remainder is 2 this indicates that the value is the 2 nd term in the sequence {i, 1, i, 1}. Therefore, the answer is 1.

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Example #5 Quotients of Two Complex Numbers Express each quotient in standard form. A. Multiply the top and bottom by the conjugate of the denominator, FOIL, and simplify. Write your final answer as a complex number of the form a + bi.

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Example #5 Quotients of Two Complex Numbers Express each quotient in standard form. B.

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Example #6 Square Roots of Negative Numbers Write each of the following as a complex number. A. B. After removing the i, make sure to place it out front as this can be confusing: This time with the i off to the side there is no confusion.

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Example #6 Square Roots of Negative Numbers Write each of the following as a complex number. C. Be sure to remove the i from each radical first!

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Example #7 Complex Solutions to a Quadratic Equation Find all solutions to the following:

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Sum & Difference of Cubes

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Example #8 Zeros of Unity Find all solutions to the following: A.

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Example #8 Zeros of Unity Find all solutions to the following: B.

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