Aim: How do we multiply and divide rational expressions?
       To multiply or divide a rational expression, you have to factor out any expression that can be factored and then multiply or divide. 

Example
3x-9/x-2*x^2-2x/x-3

1) Factor 3x-9 
3(x-3)/x-2

2) Factor x^2-2x
x(x-2)/x-3

4) Divide x-3/x-3 and x-2/x-2 
*You can cross them out as they both equal to 1

5) Multiply the remaining 3 and x on the numerator 
3x

 Excluded Value:
Find any problem that would make the denominator equal to zero.
x-2=0                           x-3=0

  +2  +2                          +3  +3

x=2                               x=3
x canont be eqaul to 2 or 3

Example
(2b^2-12b/b+5)/(b-6/b+5)

1) Keep the the fraction on the left but change the division sign and fraction 
on the right
(2b^2-12b/b+5)*(b+5/b-6)

2) Factor 2b^2- 12b
2b(b-6)

3) Divide b-6/b-6 and b+5/b+5
* Cross them out because they equal to one

4) This only leaves 2b on the numerator 
2b 
Excluded value:

b+5=0            b-6=0         

  -5  -5              +6  +6

b= 5                b=6

b cannot be equal to 5 or 6.

* Note you also use the denominator you                                                                                                      get after flipping 

How do we simplify expressions with rational exponents?

Aim: How do we simplify expressions with rational exponents?

A rational exponent contains both an integer exponent and a root (the number that must be multiplied by itself  a given times to equal a given value). The root is at the denominator and the integer exponent is at the numerator.  To simplify expressions with rational exponents, you first take the n root of the radicand and then simplify the exponent. 


First example
(16)^5/4
First, you find the 4th root of 16
4√16=2
Then, you multiply 2 by itself 5 times
(2)^5
Now this gives you the answer
32

Second example
(4^1/2)
First, you find the square root of 4
√4=2
Then you get
 4^1
Finally you get  the answer
4

Citation
http://www.mathwords.com/r/rational_exponents.htm

How do we solve radical equations?

Aim: How do we solve radical equations?

To solve radical equation, you first set the radical alone and then square both sides of the equation to get rid of the radical sign. After you get rid of the radical, you can solve it like a regular equation and solve for x. After you get the solutions, you check by plugging it into the original equation to see if the solutions make the equation true. 

For example,

√x-7+5=6

First, subtract 5 from both sides to set the radical alone.

√x-7+5=6

        -5  -5

   √x-7=1

Then, square both sides to get rid of the radical sign.

√x-7^2=1^2

*Squaring gets rid of  the radical sign

x-7=1

Now, add 7 to both sides and solve it like a regular equation.

x-7=1

  +7 +7

 x=8

Then, check by plugging it in the original equation.

√(8)-7+5=6

√1+5=6                    TRUE

1+5=6

6=6

Finally, x=8

How to factor by grouping?

Aim: How to factor by grouping? 

To factor by grouping first separate the equation into two groups and factor out the common factors from each group. Then combine the the common factors that are extracted from each group. 

Example:

x^3 - 3x^2 + 2x - 6 

1) First, separate it into two groups
(x^3 - 3x^2) + (2x - 6)

2) Then factor out the common factors from (x^3 - 3x^2)

The common factor in this group is x^2 and if you take that out      what is left is (x-3)

x^2(x-3)

3) Now factor out the common factors from  (2x - 6)

The common factor in this group is 2 and if you take that out      what is left is (x-3)

2(x-3)

*Note both groups have same expression (x-3) left after taking out the common factor and that ensures the factoring is correct. 

4) Lastly, combine the common factors extracted from each group

Final answer: (x^2+2)(x-3)

* Note you do not write (x-3) twice since is the same. 

What is common factor?

Factors that are common to two or more numbers are said to be common factors.

Citation (image):

- http://misscalculate.blogspot.com/2011/12/factoring-ax2-bx-c.html

Aim: how do we solve quadratic inequalities ? 

To solve quadratic inequality, first you can either graph or factor the inequality to see where the graph hits the x-axis. Then test different points to see which points make the inequality true. 

For example,
y< x^2-x-12  
First, factor or graph it to find where the parabola hits the x-axis 


y< x^2-x-12  
y = x^2-x-12  

Zero Product Property:
0= x^2-x-12  
0 =(x+3)(x-4)

x+3=0                         x-4=0
  -3  -3                           +4 +4   
x= -3                              x=4

Now, to find out where the points will make the inequality true, pick test points from both inside and outside the graph to test.


Inside:
(0,0)
0< (0)^2-(0)-12
0< 0-0-12
0< -12              FALSE


Outside: 
(-4,-8)
-8< (-4)^2-(-4)-12
-8< 16+4-12
-8< 20-12
-8< 8                   TRUE


Finally, the solution is: x<-2 or x>4 


Now that you know the points outside the parabola makes the graph true, you shade the outside of the parabola.

How do we use complex conjugate to divide imaginary numbers?

How do we use complex conjugate to divide imaginary numbers?

To divide imaginary numbers you write the quotient as a fraction and divide the numerator and the denominator  by the conjugate of the denominator. 

The complex conjugate of (a+bi) is (a-bi)

* Note you just change the sign in the middle. 

Example:

(1-2i)/ (3+i)

-First, find the complex conjugate of the denominator 

-Complex conjugate of (3+i) is (3-i)

-Next multiply both numerator and denominator by (3-i)

-If you foil (1-2i) and (3-i) you get 1-7i

-If you foil (3+i) and (3-i) you get 10

-Your final; answer is 1-7i/10

    

*Note i x i= i^2= -1

How to foil?

Citation (images):- http://literacy.kent.edu/Oasis/Resc/Educ/algebra.html

                          -http://www.virtualnerd.com/algebra-2/quadratics/imaginary-complex-numbers.php

How Do We Solve Imaginary Number?

We use imaginary numbers to solve problems that need the square root of negative number. 

• We use "i" as an imaginary number which equals to square root of -1.

• i is the solution to the equation i^2+1=0 or i^2=-1.

For example:

            √ -9=√ -1 √ 9
   
             =i√ 9
     
             =3i
                                                                                 

• You always put i after a real number.
• You only put i before a square root or variable.
• If a an answer contains both real number and square root or variable, you  put the i in between. (after the real number and before the square root or variable). 

         


                 
Citation (images):
-http://discoisland.wordpress.com/2010/07/15/true-love-is-less-real-than-imaginary-numbers/
-http://www.google.com/imgres?start=61&num=10&um=1&hl=en&biw=1280&bih=933&tbm=isch&tbnid=Mwd_PZKgZnK56M:&imgrefurl=http://www.mathsisfun.com/numbers/imaginary-
-http://www.coolmath.com/algebra/10-complex-numbers/01-what-are-complex-numbers-01.htm

Why do we sometimes flip the inequality symbols?

When an expression is multiplied or divided by a negative number its preference changes. For example, we have an inequality 8> 4. It means eight is greater than four. Now if we divide or multiply it by -1 it will become -8>-4. However this is incorrect as -4 is greater than -8. Therefore, we need to flip the inequality sign to make the expression mathematically true. So the correct expression is -8<-4, which is mathematically true.

     Citation (images): - http://www.studyzone.org/testprep/math4/d/numberlinel.cfm
                  -http://www.math.com/school/subject2/images/S2U3L4DP2.gif