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Algebra 2 Polynomials/Radicals Unit: 5E Notes and examples IMAGINERY/COMPLEX NUMBERS... You have been told that we canNOT evaluate , because there is a negative under the radical and the "index" is an even number (even root). Mr. McMurray What is your favorite? 2nd favorite? The truth is...WE CAN! What you should have been told is that we canNOT evaluate over the set of REAL numbers... Pattern? We CAN evaluate the square root of ANY negative number over the set of IMAGINERY numbers... guarantees >>even an integer? power. TO DO THIS WE JUST NEED 1 OR 2 SIMPLE RULES... So just how does this help us to simplify square roots that contain a negative number? So just how does this help us to simplify square roots that contain a negative number? If there is a negative under the square root... PULL AN OUT!!! If there is a negative under the square root... PULL AN OUT!!! NOTE: Any number with an "i" in it is NOT a REAL number... it is an IMAGINERY number TWO MAJOR cautions... 1. Don't just pull an out every time you see a negative under a radical... Compare these two... No "i" here because it is an ODD root! There is an "i" here because it is the SQUARE root! 2. Pull the 's out BEFORE you multiply... SIMPLIFY the following: How do I know it's not simplified already? (TWO reasons...) 1. Perfect Squares under the radical. 2. Negative under the radical. Algebra 2 Polynomials/Radicals Unit: 5E Notes and examples SIMPLIFY the following: How do I know it's not simplified already? (TWO reasons...) Mr. McMurray Every number you can possibly dream of can be written in the form of 1. Perfect Squares under the radical. 2. Negative under the radical. NOTE: which describes our FINAL set of number we call COMPLEX NUMBERS. A Complex Number comes in the form where a is the real number element, and bi is the imaginery element. is a complex number. How would I write 8 (which is obviously a real number) as a COMPLEX NUMBER? SIMPLIFY THE FOLLOWING COMPLEX NUMBERS... 1. Remove ( ) "Erase them!" 2. Combine like terms. What red flag should be waving right now? SIMPLIFY THE FOLLOWING COMPLEX NUMBERS... Any suggestions? F O I L You should hear me warning you already... F O I L 1. Remove ( ) "Erase 1st set, then Distribute the "" 2. Combine like terms. SIMPLIFY THE FOLLOWING COMPLEX NUMBERS... Why are these easier than the previous two? Because they are "conjugates" of each other. (Same numbers... different signs) All I have to multiply is FIRST and LAST! (OUTER/INNER will cancel out!) SIMPLIFY THE FOLLOWING COMPLEX NUMBERS... How do I know these aren't simplified already? ...They have a radical ("i") in the denominator... Algebra 2 Polynomials/Radicals Unit: 5E Notes and examples SIMPLIFY THE FOLLOWING COMPLEX NUMBERS... Mr. McMurray SIMPLIFY THE FOLLOWING COMPLEX NUMBERS... CONJUGATES... F O I L