Polynomial Fuctions

Hey guys,
this post will cover just some of the basics of polynomial functions. This post will focus on the divisioin of polynomials through long division and Factor Theorum.

Using Long Divison:

As many of you should alrady know how to divide, dividing with exponents isn't that much harder. So lets start with this example first. Use long division to divide 36x^3 - 6x^2 + 3x - 33 by 3x + 1 (Note: BY 3x + 1 is what they want you to divide with). So how do we do it?

1. Write down the question first
2. Like you would do for normal divison, ask yourself how many times can 3x go into 36x^3? (Answer: 12x^2)
3. After finding that out, treat the answer you found in #2 like 12x^2(3x + 1) in order to progress.
4. Subtract the answer.
5. Bring down the 3x from the question.
6. Repeat the same processas you did for 2. (how many times does 3x go into -18x^2? Answer: -6x)
7. Repeat steps 3, 4, 5, 6.
8. Once you have reached the remainder (-30R), you can now write down the division statement (Dividend) = (Divisor)(Quotient) + Remainder


Factor Theorum:

The next thing we'll look at is how to use factor theorum to solve for a polynomial (FYI: MUCH fast and WAY easier).

If you are given a question such as: Factor x^4 + 4x^3 - x^2 - 16x - 12
So how do you do it?

1. To find the first zero, find a number for x which can cause the polynomial to equal to zero (in this case -2)
2. Your new polynomial should look like (x+2)(x^3 + 2x^2 - 5x - 6) But your not done yet.
3. Repeat step one, your zero's should now become more clear. (x + 2) (x + 3) (x^2 - x - 2)
4. Finally, factor out x^2 - x - 2 (product/sum)


Well, that's about all you need to know about long division of polynomials and factor theorum. Hope this helped.

Review: Trigonometic Functions

Trigonometic Funtions Part 1



Radian Measures

• 2radians = 360 degrees, radians = 180 degrees
• to convert degrees to radians, multiply degree by (/180)
• to convert radians to degrees, multiply radians by (180/)

Special Angles

• you can use the unit circle to solve
• use special angles with unit circle

Equivalent Trigonometric Expressions

• use the cofunction identity sinx = cos (/2 - x) to determine equivalent trigonometric expressions

Compond Angle Formula

• apply equivalent trigonometric expressions to compund angle formula
• determine exact trigonometric ratios for angles expressed as sums or differences or special angles

Identities

• treat each side independantly and transform expression on one side into exact form of the other

Trigonometric Functions Part 2

Sine, Cosine, and Tan Graphs

• y=sinx and y=cosx graphs have amplitude of 1 and a period of 2 and can be transformedy=tanx has no amplitude and no maximum or minimum

Reciprocals

• reciprocals are different from it's inverse
• csc means 1/sinx
• sec means 1/cosx
• cot means 1/tanx

Mapping Rule

• g(x) = af [k(x-d)] + c
• a represents amplitude
• d represents phase shift
• c respresents vertical translation

Period

• 2/k

Amplitude

• max - min / 2

Rational Functions: Exam Review

So Exams are coming up and basically this post is going to cover some things you need to know for Rational Functions. Things such as key features and skills you need to have when dealing with rational functions.


Rational Functions

f(x) = p(x)/q(x), q(x) cannot equal 0

f(x) = 1/x this is a simple rational, it is also the reciprocal of a linear functions.
therefore...

f(x) = 1/x^2 is reciprocal of a quadratic f(x) = 1/x^3 is a reciprocal of a cubic and etc.


Special Skills Required
Things you would need to know when dealing with rationals are...
- Finding a Vertical and/or Horizontal Asymptote
- Find any y-intercepts, zero's, holes (if there are any)
- State the Domain and Range
- Graph a rational function
- Factor into factored form

Key Features
- As stated above, the denominator cannot equal 0, anything being divided by 0 causes problems.

Horizontal Asymptote (H.A.)
- if n < m, H.A. is y =0
- if n = m, H.A. is y = coefficient of x^n/ coefficient of x^m
- if n > m, there is no H.A.

Vertical Asymptote (V.A.)
- find the zero's of the denominator

Y-intercept
- Lets x=0

X-intercept
-Let y=0

Example

f(x) = x^2 - 3x - 10 / x^2 + 9x + 14

Factoring
f(x) = x^2 - 3x - 10/x^2 + 9x + 14

Quadratic formula on top and bottom
f(x) = (x-5) (x+2) /(x+7)(x+2)

the (x+2) cancels each other out
f(x) = (x-5)/(x+7) , x cannot equal 7


Horizontal Asymptotes
- n = m so...
y= coefficient of x^n/coefficient of x^m
y= 1/1
y= 1

Vertical Asymptotes
- solve for the zero's of the denominator
x= -7

Y-intercept
-let x = 0

y= (0-5)/(0+7)
y= -5/7

X-intercept
-let y = 0

0= (x-5)/(x+7)
0= x-5
x=5

Holes
x= -2


Graph


Domain and Range
D= {x| x cannot equal -7, xE R}
R= {y| y cannot equal 1,> yER}