Elementary Algebra Chapter 4
Exponents and Polynomials

From Section 4.1 and 4.2

From Section 4.3
Definition:
A number is written in scientific notation if it is written as the product of a number between 1 and 10 and an integer power of 10.

Changing from scientific notation to standard notation:
A number in scientific notation like xxx.xxx X 10n can be changed to standard notation simply by moving the decimal point in xxx.xxx
n places and omitting the power of 10. If n is positive the decimal is moved to the right and if n is negative the decimal is moved to the left.

Changing from standard notation to scientific notation:
A number in standard notation like xxx.xxx can be changed to scientific notation simply by moving the decimal point in xxx.xxx
  
n so that there is one significant digit to the left of the decimal. Count how many places the decimal was moved. If the decimal was moved n places to the right, multiply by 10-n. If the decimal was moved n places to the left, multiply by 10n.

From Section 4.4
Definition:
A term is a number, a variable or a product of numbers and variables.

Definition: Two terms are called like terms or similar terms if they have the same variables with the same exponents

Definition: The degree of a term is the sum of the exponents on the variables.

Definition: The numerical part of a term is called the coefficient of the term (sometimes called the numerical coefficient)

Definition: A polynomial is a term or a sum of terms in which all variables have whole number exponents.

Definition: The leading term of a polynomial is the term with largest degree

Definition: The coefficient of the leading term of a polynomial is called the leading coefficient of the polynomial.

Definition: The degree of a a polynomial is the degree of the leading term.

Definition: If a polynomial contains a term which is strictly numerical, it is called the constant term of the polynomial.

Definition: A polynomial consisting of a single term is called a monomial.

Definition: A polynomial consisting of two terms is called a binomial.

Definition: A polynomial consisting of three terms is called a trinomial.

From Section 4.5
Process: To add like terms we add their coefficients and keep the same variables with the same exponents

Process: To add polynomials we add like terms.

FACT:   The sum of polynomials is a polynomial.

From Section 4.6
Process: To multiply two monomials, multiply the numerical factors and then multiply the variable factors.

Process: To multiply a polynomial by a monomial, multiply each term of the polynomial by the monomial.

FACT:   Multiplying a polynomial by a monomial is an application of the distributive property.

FACT:   The product of a polynomial and a monomial is a polynomial.

Process: To multiply a binomial by a binomial, multiply each term of the second binomial by each term of the first binomial and combine like terms.

Process: To multiply two polynomials, multiply each term of the second polynomial by each term of the first polynomial and combine like terms.

FACT:   The product of two polynomials is a polynomial.

Some Special Products:
The square of a sum:                                       (x + y)2 = x2 + 2xy + y2
The square of a difference:                            (x - y)2 = x2 - 2xy + y2
The product of a sum and difference:         (x + y)(x - y) = x2 -  y2

From Section 4.7 and Section 4.8
Process: To divide a polynomial by a monomial we divide each term of the polynomial by the monomial.

FACT:   The quotient of a polynomial divided by a monomial is a polynomial.

Process: To divide a polynomial by a polynomial we use a process similar to long division from arithmetic.

FACT:   The quotient of a polynomial divided by a polynomial is a polynomial.

Process: To divide a polynomial by a monomial we divide each term of the polynomial by the monomial.

FACT:   When a polynomial P is divided by a polynomial D, it is always possible to obtain a polynomial Q called the quotient and a polynomial R called the remainder so that P = DQ + R. Moreover, the degree of R is less than the degree of D.