OP - Operations with Polynomials Module Overview

Operations with Polynomials Module Overview

Introduction

image of equation f(x)=4x to the 4th+3x+7
where degree is to the 4th
leading coefficient is 4
variable is x
constant is 7 In mathematics, a polynomial is an expression consisting of variables and coefficients, where the exponents are non-negative integers

Polynomials can also be classified by their degree, which is the sum of the variable exponents in any one term.

Polynomial expressions have the following operations: addition, subtraction, multiplication, division.

Polynomials have many applications; here are a few examples of some applications:

Planetary Motion : Johannes Kepler (1571-1630) discovered that the orbits of planets in our solar system follow paths around the sun that are ellipses. Kepler discovered several laws about planetary motion, one of which is described below.
Pierre Fermat (1601-1665) conjectured that every number of the form, $LaTeX: F_n\:=\:2^{2^n}\:+\:\:1$ $F_n\:=\:2^{2^n}\:+\:\:1$ , where n is a nonnegative integer, is prime. These numbers are called Fermat numbers in his honor.
The following formula gives the monthly cost M of a mortgage (a loan taken out with real estate as collateral) if the principal (the amount borrowed) is p dollars, which is to be totally repaid in n months, and the monthly interest rate (as a decimal) is r: $LaTeX: M=\frac{rp}{1-\left(1+r\right)^{-n}}$ $M=\frac{rp}{1-\left(1+r\right)^{-n}}$
- Constructing a Pyramid
For a class project, you and your classmates are constructing two pyramids using aluminum cans. One of your pyramids will have a triangular base and the other will have a square base as shown below.The formula for the total number of cans C in a triangular-base pyramid with n layers is . For a square-base pyramid, it is $LaTeX: C=\frac{1}{3}n^3+\frac{1}{2}n^2+\frac{1}{6}$ $C=\frac{1}{3}n^3+\frac{1}{2}n^2+\frac{1}{6}$ .
- Economic Models
Economic models are used by both large and small businesses. Suppose you work in the marketing department of a worldwide manufacturer and your company has recently released a new product on the market. The demand function p shown at the right for the new product is $p=34-2x^2$ where x is the number of units produced in millions.

Essential Questions

What are Polynomials?
How are Polynomials classified by type and degree?
How are Polynomials added, subtracted, multiplied, and divided?
What is Pascal's Triangle and how it is used in multiplying Polynomials?
How are Polynomials expressions divided using Synthetic Division?
How are Polynomials expressions divided using Long Division?
What are the Compositions of Functions?
What are the Inverses of Functions?
How are Inverses of Functions verified?

Operations with Polynomials Key Terms

Polynomial - The sum or difference of two or more monomials.

Constant - A term with degree 0 (a number alone, with no variable).

Monomial - An algebraic expression that is a constant, a variable, or a product of a constant and one or more variables (also called "terms").

Binomial - The sum or difference of two monomials.

Trinomial - The sum or difference of three monomials.

Integers - Positive, negative and zero whole numbers (no fractions or decimals).

Like Terms - Terms having the exact same variable(s) and exponent(s).

Coefficient - Number factor; number in front of the variable.

Linear - A 1^st power polynomial.

Quadratic - A 2^nd power polynomial.

Cubic - A 3^rd power polynomial.

Quartic - A 4^th power polynomial.

Pascal's Triangle - A number triangle with numbers arranged in staggered rows such that equation image indicator $LaTeX: a_{nr}=\frac{n!}{r!\left(n-r\right)!}=\frac{n}{r}$ $a_{nr}=\frac{n!}{r!\left(n-r\right)!}=\frac{n}{r}$ , where $LaTeX: \left(\frac{n}{r}\right)$ $\left(\frac{n}{r}\right)$ is a binomial coefficient.

Synthetic Division - Is a shorthand, or shortcut, method of polynomial division in the special case of dividing by a linear factor - and it only works in this case.

Long Division - Is an algorithm for dividing a polynomial by another polynomial of the same or lower degree.

Compositions of Functions - A composition of a function is the point wise application of one function to the result of another to produce a third function. Intuitively, composing two functions is a chaining process in which the output of the inner function becomes the input of the outer function.

Inverses of Functions - An inverse of a function is a function that undoes the action of another function. A function g is the inverse of a function f if whenever y = f(x) then x = g(y). In other words, applying f and then g is the same thing as doing nothing. We can write this in terms of the composition of f and g as g(f(x)) = x. A function f has an inverse function only if for every y in its range there is only one value of x in its domain for which f(x) = y. This inverse function is unique and is frequently denoted by f^-1 and called "f inverse."

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