Warm-Up

•Simplify the following terms:

TEST

•Our Ch. 9 Test will be on 5/29/14

Complex Number Operations

Learning Targets

•Adding Complex Numbers•Multiplying Complex Numbers•Rules for Adding and Multiplying

Conjugates

Addition of Complex Numbers• When adding imaginary numbers we combine

like terms

Ex:

12+12 𝑖

Multiplication of Complex Numbers

• When multiplying complex numbers we will distribute the factors throughout

Ex:

10−20 𝑖

Multiplying Notes•Be careful to notice that when multiplying

we will often end with an imaginary term to the second power.

•These terms will always simplify to their opposite value.

•Ex:

***

You Try𝑎 .14 𝑖+10−2 𝑖

𝑏 . 𝑖− 5 𝑖+3−17 𝑖− 3

𝑐 . (6− 5 𝑖 )+(6+5 𝑖 )

𝑑 . (4 𝑥− 16 𝑖𝑥 )+51+16 𝑖𝑥

12 𝑖+10

−21 𝑖

12

4 𝑥+51

You Try𝑎 .− 2𝑖 (14 𝑖+10)

𝑏 . (3+2 𝑖 ) (9 −14 𝑖 )

𝑐 . (6 − 5 𝑖 )2

𝑑 . (3+7 𝑖 ) (3−7 𝑖 )

28− 20 𝑖

55−24 𝑖

1 1− 60 𝑖

58

Conjugate Operations

•Complex Conjugate operations are needed in order to factor quadratics and determine their complex roots.

•There are two main operations that we need to know about

Sum of Complex Conjugates

•The sum of our conjugates will always result in twice the value of our real terms

Multiplication of Complex Conjugates

•Multiplying the conjugates will always result in the sum of our a terms squared and b terms squared

Why is the Conjugate Important•The conjugate is important because our

non real roots of polynomials always come in pairs

Our pairs of complex numbers will always be conjugates

Conjugate cont.

•So if we multiply our roots we should get our polynomial in standard form

Now we can begin to divide polynomials•In order to divide polynomials we have to

be able to determine one of its factors

•Once a factor is known we can begin to divide it throughout the standard form of the polynomial and simplify it

•If the factor used is indeed a root our remainder will be zero

Division Cont.

•We can then repeat the process until we are only left with all of the roots of the polynomial

•This process allows us to transform a polynomial from Standard Form to Factored Form

Types of Division

•There are two methods that we can use to divide polynomials

▫Long Division▫Synthetic Division (preferred method)

First divide 3 into 6 or x into x2

Now divide 3 into 5 or x into 11x

Long Division If the divisor has more than one term, perform long division. You do the same steps with polynomial division as with integers. Let's do two problems, one with integers you know how to do and one with polynomials and copy the steps.

32 698 x - 3 x2 + 8x - 5

2 x

64 x2 – 3x

Now multiply by the divisor and put the answer below.

Subtract (which changes the sign of each term in the polynomial)

5 11x

Bring down the next number or

term8

- 5

1 + 11Multiply and

put below

3211x - 33

subtract

2628

This is the remainder

3

28

x

Remainder added here over divisor

So we found the answer to the problem x2 + 8x – 5 x – 3 or the problem written another way:

3

582

x

xx

List all coefficients (numbers in front of x's) and the constant along the top. If a term is missing, put in a 0.

You want to divide the factor into the polynomial so set divisor = 0 and solve for first number.

Let's try a problem where we factor the polynomial completely given one of its factors.

502584 23 xxx

- 2 4 8 -25 -50

4

Bring first number down below lineMultiply these and

put answer above line

in next column

- 8 Add these up

0Multiply these and

put answer above line

in next column

0 Add these up

- 25

50

0

Multiply these and

put answer above line

in next column

Add these up

No remainder so x + 2 IS a factor because it

divided in evenlyPut variables back in (one x was divided out in process so first number is one less power than original problem).

x2 + x

So the answer is the divisor times the quotient:

2542 2 xx

2 :factor x

You could check this by multiplying them out and

getting original polynomial

Comparison Between Synthetic and Long Division

•Why Synthetic Division Works

Example:

•Is the factor a root of:

You try:

•Is the factor a root of:

You try:

•Is the factor a root of:

You try:

•Is the factor a root of:

For Tonight

•Worksheet