warm-up
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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. - PowerPoint PPT PresentationTRANSCRIPT
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 𝑖58
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 - 52 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
328
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:
3582
xxx
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
0Multiply 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