Polynomial+Operations

The basic polynomial operations include the following operations: (i got this following information at http://zone.ni.com/reference/en-XX/help/371361B-01/lvanlsconcepts/basic_polynomial_operations/) You can find the product of two polynomials by using the familiar axioms of addition and multiplication and the first law of exponents. Here is an example of the following: ( 3x-2 )(5x^2 - x^3 + 4x) = 3x (5x^4 - x^3 + 4x) - 2(5x^4 - x^3 + 4x) = 15x^5 - 3x^4 + 12x^2 - 10x^4 + 2x^3 - 8x = 15x^5 - 13x^4 + 2x^3 + 12x^2 - 8x
 * Finding the order of a polynomial
 * Evaluating a polynomial
 * Adding, subtracting, multiplying, or dividing polynomials
 * Determining the composition of a polynomial
 * Determining the greatest common divisor of two polynomials
 * Determining the least common multiple of two polynomials
 * Calculating the derivative of a polynomial
 * Integrating a polynomial
 * Finding the number of real roots of a real polynomial
 * __Multiplying a Polynomial__**

For multiplying a polynomial, you can use the F.O.I.L. technique. The F.O.I.L. technique is one that makes multiplying a polynoimal much easier. It stands for First Outer Inner Last Here is an example of the FOIL technique being used.

(x+3)(x-2) =x^2 - 2x + 3x - 6 =x^2 + x - 6 This would be your final answer.

Here is the link to a very good, and well thought out video on how to factor out polynomials, as well as trinomials and binomials. http://www.youtube.com/watch?v=uoEoWzHXaJ8
 * __Factoring Polynomials__**

__**Here are some very useful defintions and sayings about polynomials.** Here is an example of the following: 2x^2 + 6x/ 4x^3 = (x + 3)/ 2x^2 This is so because 2x is your greatest monomial factor, or to put in normal english, it can be used in every other factor. 2x will factor out 2x^2, 6x, and 4x^3. so your final answer would be as followed: (x + 3) --over-- 2x^2
 * G.M.F. =**__ **Greatest Monomial Factor**

25 is a perfect square because it can be broken down into 5 * 5. 36 is a perfect square because it can be broken down into 6 * 6. x^2 is a perfect square because it can be broken down into X * X To find a perfect square's identical square, cut the power in half. (a * b)^n = a^n * b^n b^n * b^m = b^(n+m) 3^2 * 3^5 = 3^7 (b^m)^n = b^m*n (3^2)^5 = 3^10 (a+b)(c+d) = ac + ad + bc + bd a 2 - b 2 = (a+b)(a-b) (Difference of squares) a 3 + b 3 = (a + b)(a 2 + ab + b 2) (Sum and Difference of Cubes)
 * __The term //To Factor// means this:__** To re-write a polynomial as a product.
 * __The Difference of Squares is as followed:__** A perfect square is a number that can be broken down into two (2) identical factors.
 * Anything raised to an even power is a perfect square***
 * __The Power of Product:__**
 * __The Product of Powers__**
 * __The Power of Power__**
 * __The following equations and examples can be found on Math.com__**
 * (a+b) 2 = a 2 + 2ab + b 2**

**__The Breakdown of a Polynomial__** 2x^2 + 4x + 2 2x^2 = The Quadratic Term 4x = The Linear Term 2 = Constant

To factor out a polynomial, you must use all of the following information listed above. For example, to factor out the polynomial x^2 - 5x - 6, The only thing that factoring means is pretty much breaking it down. so to break down x^2, it would be (x*x) To get the final answer, you have to break it down to where the first (quadratic term) will equal x^2, the second (linear term) will equal -5x, and the third (constant term) will equal -6. So to work this out, we need to break it down like this. (x-6)(x+1) This is the answer because x*x equals x^2, 1x - 6x equals -5x, and -6 * 1 equals -6.
 * __Factoring A Polynomial__**

Using the previous information, try to solve the following questions.
 * __Practice Problems__**

1) (x+3)(x+2)

2) (2x-3)(2x+7)

3) (x^2+3)(x^2-3)

4) (15x^2+4x+2)(10x^2+-4x-2)

5) (x+5)(x-5)

"(unknown)." __National Instruments__. 20 Nov. 2008 . "(unknown)." __Math.com__. Polynomial Identities. 20 Nov. 2008 .)