**Elementary
Algebra Chapter 4
Exponents and Polynomials**

**From
Section 4.1 and 4.2**

**From
Section 4.3
**

**Changing
from scientific notation to standard notation:
**A number in scientific notation like xxx.xxx X 10

**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

**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} = x^{2} + 2xy + y^{2}**

The square of a difference: (x - y)^{2} = x^{2} - 2xy + y^{2}

The product of a sum and difference: (x + y)(x - y) = x^{2} - y^{2}

**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.