The Fundamental Theorem of Algebra Lesson 4.4

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

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

Complex Numbers Combine real numbers with imaginary numbers Examples a + bi Examples Real part Imaginary part

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

Operations on Complex Numbers Complex numbers can be combined with addition subtraction multiplication division Consider

Operations on Complex Numbers Division technique Multiply numerator and denominator by the conjugate of the denominator

Complex Numbers on the Calculator Possible result Reset mode Complex format to Rectangular Now calculator does desired result

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

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

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

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

Example Consider the solution to Note the graph No intersections with x-axis Using the solve and csolve functions

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

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

Assignment Lesson 4.4 Page 279 Exercises 1 – 87 EOO