Complex Numbers

A number in the form:

represents a complex number in Standard Form

In standard form a is called the real part of the complex number and bi (where b is a real number) is called the imaginary part 

If b=0, the number a+bi=a is a real number. If b does not equal 0, the number a+bi is called an imaginary number. A number of the form bi, where b does not equal 0, is called a pure imaginary number.

Equality of two complex numbers:

Operations with Complex Numbers:

Sum:

Difference:

Additive inverse:




So,

Complex Conjugates:
A pair of complex numbers of the form a+bi and a-bi that have a real number as their product

To find the quotient of a+bi and c+di where c and d are not both zero, multiply the numerator and denominator by the conjugate of the denominator to obtain

=

The Fundamental Theorem of Algebra

Any polynomial of degree "n" has "n" roots

however, the roots can be complex (imaginary)

This function has 4 roots, since the highest exponent is 4


A function can also be rewritten using its roots:

This function has 2 roots because of the 2nd degree exponent

Factor into :
The roots are 4 and -4, so the equation can be rewritten using the roots to read:

Imaginary roots:
If a function's highest degree is 5, there will always be 5 roots. However, some of these roots may be imaginary, or, complex.

If a root is complex, its conjugate must also be complex.

Therefore, complex roots come in pairs and there will always be an even amount of them.

For example, the following equation has 2 roots. They are both imaginary though, because:

Remainder, Rational and Factor Theorems

The Remainder Theorem

The Remainder Theorem is basically a method to evaluate a polynomial at a given value.
For example:

polynomial p(x) = 5x⁴ + 2x³ + 4x

If we want to evaluate this function at x = 2, we can do two things.
One is the simple plug the value 2 in and evaluate, but the Remainder Theorem can also work.
First step in using the Remainder Theorem is synthetic division. 
So for this example and x = 2 we have:

   2  |  5   2    0     4    0
       |      10   24  48  104
          5  12   24  52  104

The remainder is 104.  This should match what we would get if we plugged x = 2 into the original polynomial.
Lets check:

5 * 16 + 2 * 8 + 4 * 2 = 104

An important reason to use the Remainder Theorem is evaluate a value of x for the root(s) of a polynomial.
If for a given x the remainder is 0 then x is a root.

The Rational Root Theorem

The Rational Root Theorem states that it is possible to find all possible rational roots of a function by determining:

+          factors of the constant       
      factors of the leading coefficient 

     
For example, consider the equation:

The possible roots would look like

 +    1,3   

        1

meaning that all the possible roots are: 1, -1, 3, -3

 

The Factor Theorem

The factor theorem is essentially the remainder theorem in reverse. If a polynomial is synthetically divided by x=a and a remainder of zero is found then x=a is a zero which is essentially the Remainder Theorem, but what is key to the Factor Theorem is this also means (x-a) is a factor of the polynomial.

Helpful sites:

http://www.purplemath.com/modules/remaindr.htm

http://www.purplemath.com/modules/rtnlroot.htm

http://www.purplemath.com/modules/factrthm.htm




Jack Kelly

Dividing Polynomials (synthetic division and long division)

Synthetic Division

1. Place the coefficients in descending order on inside

2. Place number (possible zero or the zero given) on the outside

pay attention- if there is a missing term don't skip it replace it with zero... 

final answer written as 

 

If you are not given the number(possible zero) to divide by, then you must find the possible numbers yourself.

To find this zero you take the factors of the last term, the constant, and divide them by the factors of the first term, the leading coefficient. (rational root theorem)

 Factor out an x 

x=0

 a shortcut to find a zero: when all the coefficients add to zero, the zero is 1.If this does not work then it is pretty much trial and error based off of the rational root theorem

 x=1

x=1 again

now you are left with 

which can be factored to

x=1, x= -2

the zeros altogether are x=1 mult. of 3 and x= -2

Long Division:

final answer written as...

*synthetic division is when you are dividing by a first degree polynomial and long division can be used for a first degree or any higher degree*

videos that might help:
synthetic division-
http://www.youtube.com/watch?v=bZoMz1Cy1T4
http://www.youtube.com/watch?v=1byR9UEQJN0

long division-
http://www.youtube.com/watch?v=l6_ghhd7kwQ
http://www.youtube.com/watch?v=ok4k6HDCuIk

Completing the Square

A polynomial function is of the form:



Important to note:

-the value of n must be a nonnegative integer (It must be a whole number, it is equal to zero, or it is a positive integer)

-the coefficients are These are real numbers.

The degree of a polynomial function is the highest value for n where is not equal to zero.

Determining the Degree of the Polynomial Function

Degree: 0
Name: constant

Degree: 1
Name: linear

Degree: 2
Name: quadratic

Degree: 3
Name: cubic

Completing the Square

Complete the Square:

Step One:

Put equation into Standard Form

Step Two:

Ignore the 7 for now and work with the equation...

Step Three:

Make a perfect square by diving 8 by 2 and squaring the new number and adding it to the equation. 

Step Four:

Simplify

Step Five:

You have added 16 to your original equation and therefore must subtract 16 to even it out. 

Step Six:

Add the 7 and simplify

Step Seven:

Solve for x.

Example:

Helpful Videos:

http://www.youtube.com/watch?v=gzm-uhj06q8&feature=related

http://www.youtube.com/watch?v=xGOQYTo9AKY

Zeroes of Functions and Multiplicity

     The first step to finding the zeroes of a function is to replace y or f(x) with 0. From there, you can use different methods to solve, depending on the degree of the polynomial.  For a quadratic, you can factor it or use the quadratic formula.  If the equation has a degree larger than two, you can use synthetic division, factoring by grouping, etc.
     ex:

   

However, the answer isn't just "x= -2", because there were two of them (a repeated zero).  To compensate for that, we say that the solution is -2, with a multiplicity of 2 (because it was there twice).
     A factor of the expression below yields a repeated zero x=a of multiplicity k. 

• if k is odd, the graph crosses the x-axis at (a , 0).
• if k is even, the graph only touches the x-axis at (a , 0).

     So, in the previous example, the graph would touch the x-axis at x=-2 because the solution had an even multiplicity. 

Things to remember about zeroes:

• a polynomial with degree n has at most n zeroes, including the multiplicities
• some functions do not ever cross the x-axis because their solutions are imaginary

If you need further explanation on finding the zeroes and determining their multiplicity, this video is really helpful.

More helpful links:

http://www.purplemath.com/modules/polyends2.htm 

http://primes.utm.edu/glossary/xpage/Zero_.html

End Behavior of Polynomial Functions

Laws of the End Behavior of a Polynomial Function

End Behavior of Polynomial Functions: 

The characteristic of a continuous function at extreme points. 

Polynomial Function Expression:
a_nxⁿ+a_n-1x^n-1+a_n-2x^n-2… +a₂x²+a₁x¹+a₀

Odd Exponents (n): 

When the leading coefficient is positive (a_n>0), the graph falls to the left and rises to the right: 

f(x) → ∞ 

as x → ∞

&

f(x) → -∞ 

as x → -∞

When the leading coefficient is negative (a_n<0), the graph rises to the left and falls to the rights. 

f(x) → ∞ 

as x → -∞

&

f(x) → -∞ 

as x → ∞

Even Exponents (n): 

When the leading coefficient is positive (a_n>0), the graph rises to the left and right.

f(x) → ∞ 

as x → -∞

&

f(x) → ∞ 

as x → ∞

When the leading coefficient is negative (a_n<0), the graph falls to the left and right.

f(x) → -∞ 

as x → -∞

&

f(x) → -∞ 

as x → ∞