Polynomial Functions

A polynomial function is one that can be written in the form

a_nx + a_n-1x^n-1+…+a₃x³+a₂x² + a₁x+ a₀

_{Where n is an non negative integer, x is a variable, and each of}  a₀,a₁,a₂,…a_n is a constant called a coefficient. 

 Rational Functions:
Find the domain of each rational function
1) f(x) = 1/x²
The domain of  f(x) = 1/x² is the set of all real numbers except x = 0 because the denominator is 0 when x=0, making the fraction undefined
2) g(x) = x² + 3x + 1/x²-x-6
The domain of g(x) = x² + 3x + 1/x²-x-6 is the set of all real numbers except the solutions of x²-x-6=0. Because x²-x-6 factors into (x+2)(x-3), the solutions to x²-x-6=0 are x=-2 and x=3. Therefore, the domain of g is the set of all real numbers except x = -2 and x=3



 Real Functions:


Synthetic Division:

Inequalities

Inequalities
-Range of values for a solution.

Solving a compound linear inequality:

Example:

2<3x+5<2x+11

A solution of 2<3x+5<2x+11 is any number that is a solution of both of the following inequalities.
2<3x+5 and 3x+5 <2x +11
each of these inequalities can be solved by the principles listed above.

2 - 5 <3x       3x - 2x< 11- 5
-3<3x             x<6
-1<x

the solutions are all real numbers that satisfy both -1<x and x<6, that is -1<x<6. Therefore, the solutions are the numbers in the interval [-1,6).

Example:
 4 < 3-5x<18
When a variable appears only in the middle of a compound inequality the process can be streamlines by performing any operation on each part of the compound inequality.
4<3-5x<18
1<-5x<15
-1/5> x > -3
Intervals are usually written from the smaller to the larger, so the solution to the compound inequality is -3<x<-1/5
The solution of a compound inequality is the interval (-3,-1/5)

Kahn Academy:
http://www.khanacademy.org/video/algebra--solving-inequalities?playlist=Algebra
http://www.khanacademy.org/video/quadratic-inequalities?playlist=Algebra

Arithmetic and Geometric Sequences

-An arithmetic sequence has a constant difference and can be expressed as a line.

-A geometric pattern is a not linear, it is exponential.

Arithmetic Sequence:

In an arithmetic sequence {U_n}

U_n=U_n-1+d

And for some constant d and all n>2

Example:

If {U_n} is an arithmetic sequence with U₁= 3 and U₂ = 4.5 as its first two terms,

a)      a)  find the common difference

b)     b)  write the sequence as a recursive function

c)     c) give the first seven terms of the sequence

Solution:

A)     a) the sequence is arithmetic and has a common difference of U₂-U₁ = 4.5 - 3 = 1.5

B)      b) the recursive function that describes the sequence is U₁ = 3 and U_n-1 + 1.5 for n>2

C)     c) the first seven terms are 3, 4.5, 6, 7.5, 9, 10.5 and 12

Geometric Sequence:

In a geometric sequence {U_n}

U_n=RU_n-1

For some U₁ and some nonzero constant r and all n>2

 Example:

Is the following sequence geometric? If so, what is the common ratio? Write each sequence as a recursive function.

{3,9,27,81...}

The sequence is geometric with a common ratio of 3.

U₂/U₁ = 9/3  = 3

U₃/U₂ = 27/9 = 3

U₄/U₃ = 81/27 = 3

Because each term is obtained by multiplying the precious term by 3, the sequence may be denoted as a recursive function

U₁ = 3 and U_n=3U_n-1 for n>2

Kahn Academy:
http://www.khanacademy.org/video/sequences-and-series--part-1?playlist=Calculus

Using Quadratic Formula and Completing the Square

Ax^2 + Bx + C = 0

x^2 +( b/a)x = -(c/a)

1.    1.    x^2 +(b/a) + (b/a)^2 = (b/2a)^2 – (c/a)

2.    2,    (x + (b/2a)^2) = (b/2a)^2 – (c/a)

3.   3.    x +(b/2a) = + or -   (b^2/4a^2)-(c/a)

4.   4.    x = + or - Square Root      b^2/4a^2 – c/a

5.   5.    x = -b/2a + or - Square Root    b^2-4ac/4a^2

6.  6.     x = -b/2a + or -  Square Root     b^2-4ac/2a

7.  7.     x = -b + or - Square Root    b^2 – 4ac/2a  

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