Download Algebra 2 Polynomials/Radicals Unit: 5E Notes and examples

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