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College algebra CHAPTER 3

COMPLEX NUMBERS

Identify the real and imaginary parts of the complex numbers.

Adding complex numbers:

Multiplying complex numbers:


Dividing complex numbers: Conjugate of

3 +2i is 3 -2i
-3 - i is -3 + i

Find for each:

1. Horizontal moves (x)
2. Vertical moves(y)
3. Vertex
4. Up or down positive or negative
5. write the probable equation
6. What is the maximum/minimum value of the function ?

Completing the Square Warm Up:

1. Look at the previous and fill in the blanks without factoring:


so,


Vertex formula

Zero factor property
Square root property (one x)
Quadratic formula ( two x ’s)

Discriminant --------

 

 

for r for c for v

Solving quadratic inequalities

 

This is when infinitely many
solutions occur

This is when a no solution occurs

 

 

 

Check with x = 0

 

Check with x = 0

Let's look at this one graphically

Let's look at this one graphically

THE SUM OF TWO POSITIVE NUMBERS

So we basically have the following:

a) If one number is x , then the other number is __________________

b) The restriction on x is:



c) The function that represents the product of the two numbers is:

d) Find the maximum for the function and the two numbers.

A farmer wishes to enclose a rectangular region bordering a river with fencing, as shown in the diagram.
Suppose that x represents the length of each of the three parallel pieces of fencing. She has 600 feet of
fencing available.

i) Finish the equation y + = 600


Solve for y =_________________
a) If the length of each of the three parallel pieces is x,

then the length of the remaining side in terms of x is:

b) The restriction on x is:

c) The function that represents the area of the fenced region is: A= L times W = ______ times __________

d) Find the maximum area and the dimensions .

e) What would the dimensions be for an area of 22,500 square ft?

A piece of sheet metal is 2.5 times as long as it is wide. It is to be made into a box with an open top by cutting 3-
inch squares from each corner and folding up the sides. Let x represent the width of the original piece of sheet
metal.

a) The restriction on x is x > 6 why?

b) Determine a function that represents the volume:

c) For what value of x will the volume of the box be 600?

e) For what value of x will the volume of the box be 800?

d) For what values of x will the volume of the box be between 600 and 800?

A kite is flying on 50 ft of string. How high is it above the ground if its height is 10 feet more than the
horizontal distance from the person flying it? Assume the string is being held at ground level.


When Respect Brings Success charges $600 for a seminar on management techniques, it attracts 1000 people.
For each decrease of $20 in the charge, an additional 100 people will attend the seminar. Let x represent the
number of $20 decreases in the charge.

a) Determine a revenue function R that will give revenue generated as a function of x, the number of $20
decreases.


Revenue= Price X Number Sold

b) Find the value of x that maximizes the revenue. What should the company charge to maximize the revenue?

c) What is the maximum revenue the company can generate?

Graphing Polynomials

Draw the end behavior

+ leading coefficient , odd degree +leading coefficient, even degree
- leading coefficient, odd degree - leading coefficient, even degree

EXTREMA

Local maxima--- the highest point at a peak.
Local Minima----the lowest point at a valley


Turning Point
Degree give the number of possible x-intercepts

Copy the graph from your calculator and locate and label the local maxima and minima.

Absolute Maxima--- the highest point at a peak is also the highest point on the graph.
Absolute Minima----the lowest point at a valley is also the lowest point on the graph.

If there is an absolute maximum and/or minimum , then label them.
What are the x-intercepts?

Dividing by polynomials

Steps :
1)
2)
3)
4)

Synthetic division only when dividing with the form of x – r.

Find P(3)

divide

Find P(-2)

divide

Find P(0.5)

divide
Remainder Theorem:
If a polynomial P (x) is divided by x-k, the remainder is equal to P(k).

Find P(-2)

divide

if x=-2 then
if x=-3 then

We call -2 and -3 Zeros of

Determine if 3 is a zero of , use synthetic and the remainder theorem

This means that if 3 is a zero of

Then the factored form of would partially be



What would go into the parenthesis ?



How would you find the remaining portion?

If one of the zeros is -5, then what are the others?

Intermediate Value Theorem

If one value of the polynomial is negative and one value is positive, then there is zero
between those two values.

Review:

If one of the zeros is 1-i, then what are the others?

The degree of the polynomial is __________. Which means there are _________
possible distinct roots( they could repeat).

If one of the zeros is-5i, then what are the
others?

The degree of the polynomial is __________. Which means there are _________
possible distinct roots( they could repeat).

Find a polynomial of degree 3 that has the following conditions:
(don't forget the constant)
1) Zeros of -3, -1, and 4

2) Zeros of -3, -1, and 4. P(2)=5
1) Zeros of 2 and 3i 2) Zeros of 2 and 3i . P(0)= - 36
1) Zeros of , and 3

Finding Possible zeros.

Multiply
Zeros are

If one of the zeros is-5i, then what are the others?

What are the possible Zeros?
Last number____. Factors of last number__________________________
First number____. Factors of first number__________________________
 

Factor


P(x) ______________________

Graphing Polynomials By Hand

End behavior? ___________
Zeros?___________

End behavior? ___________
Zeros?___________

Solve. Find some of the zeros graphically?

Solve. Find some of the zeros graphically?

Graphing Polynomials By Hand and answer the questions

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