Warm Up Divide the complex number3 – 2i 1 + i Multiply the complex number(3 -2i)(1+i)

Math IV Lesson11 Complex numbers 2.5Essential Question: Standard: MM4A4. Students will investigate functions. a. Compare and contrast properties of functions within and across the following types: linear, quadratic, polynomial, power, rational, exponential, logarithmic, trigonometric, and piecewise.

New Vocabulary

The degree of a polynomial with one variable is the largest exponent of that variable.

Root: where the polynomial is equal to zero.

A quadratic factor with no real zeros is said to be prime.

The degree of a polynomial with one variable is the largest exponent of that variable.

RootsA root is where the polynomial is equal to

zero

So, a polynomial of degree 3 will have 3 roots (places where the polynomial is equal to zero). A polynomial of degree 4 will have 4 roots. And so on.

Example: what are the roots of x2 - 9?

x2 - 9 has a degree of 2, so there will be 2 roots.

Let us solve it. We want it to be equal to zero:

x2 - 9 = 0

First move the -9 to the other side:

x2 = +9

Then take the square root of both sides:

x = ±3

So the roots are -3 and +3

A polynomial can be rewritten like this:

• The factors like (x-r1) are called Linear Factors, because they make a line when you plot them.

Polynomials can have complex roots

• A quadratic factor with no real zeros is said to be prime.

Example: X2 + 1 X2 + 1 = 0 X2 = -1X = ± = ± i

Complex roots always come in pairs

Example: x2-x+1Had these roots:

0.5 - 0.866i and 0.5 + 0.866i

You can either have: No complex roots2complex roots4 complex roots6 complex roots…

Factoring a polynomial

Factor f(x) = x4 –x2 -20x4 –x2 -20 = (x2 – 5)(x2 +4) = (x2 – 5)(x +2i) (x –2i) = (x + ) (x - ) (x +2i) (x –2i)

Use the quadratic formula to solve find the zeros ofF(x) = x2 -12x + 26

QUADRATIC FORMULA

Homework

• P144 # 1-4, 11-19 odd