midnight of August 28th










midnight of August 28th

Exponents and Polynomials

Section 4.1 Exponents and Their Properties

An exponential expression is an expression that has a base and an exponent
baseexponent. The base can be a constant, variable, or algebraic expression .

We evaluate this expression by writing it in expanded form and using repeated
multiplication. The base is repeated as a factor for the number of times represented by the
exponent.

When the base contains variables, we can utilize the properties of exponents to help us
simplify the exponential expression.

Properties of Integer Exponents
For any base a and b and nonnegative integer exponents n and m,

for n factors of a

where

Section 4.8 Negative Exponents and Scientific Notation

When an exponential expression contains negative integer exponents, it is standard, in
most cases, to transform the expression to have only positive integer exponents. To do
this, we need some more rules.

More Properties of Integer Exponents
For any base a and b and nonnegative integer exponents n,

where

where

Scientific Notation
A number is said to be in scientific notation when expressed in the form N ×10m , where
1≤ lNl < 10 and m is any integer. The number N is called the mantissa and m the

characteristic of N ×10m .

Scientific notation is used to express very large and very small numbers. When solving a
problem where one of the inputs is in scientific notation, it is customary to express the
answer in scientific notation.

To express a large number in scientific notation
1. Move the decimal point to the left until there is only one digit to the left of the
decimal point.
2. Drop zeros from the right until you reach the first non-zero number. The resulting
number is the mantissa, N.
3. The characteristic, m, is the number of decimal places the decimal point was
moved to the left. In this case, m is a positive integer.
4. Express the result as N ×10m .

To express a small number in scientific notation
1. Move the decimal point to the right until there is only one non-zero digit to the
left of the decimal point.
2. Drop all zeros to the left of the new decimal place. The resulting number is the
mantissa, N.
3. The characteristic is the number of decimal places the decimal point was moved
to the right. In this case, m is a negative integer.
4. Express the result as N ×10-m .

To convert a number in scientific notation back to decimal notation also requires moving
the decimal point.
• If the characteristic m is positive, move the decimal point m decimal places to the
right. Place zeros in all place- values that do not contain numbers in them.
• If the characteristic m is negative, move the decimal point m decimal places to the
left. Place zeros in all place-values that do not contain numbers in them.

Shortcuts
• If the characteristic m is positive, add enough zeros to the right of the mantissa so
that m digits occur after the decimal point, then move the decimal point to the end
of the number.
• If the characteristic m is negative, add m zeros to the left of the mantissa and place
a decimal point between the first two zeros.

Section 4.2 & Section 4.6 (Exercises 1-22) Polynomials

Recall that the terms of an expression are its addends (i.e. quantities separated by plus or
minus signs). There are two types of terms.
Constant terms, which represent only one number, and
Variable terms, which represent different numbers

The numerical coefficient (or coefficient) of a term is its numerical factor.

When all of the terms in an algebraic expression contain the properties of:
1. Real number coefficients
2. Exponents on all variables are whole numbers
3. Variable terms do not appear in a denominator or under a radical
Then, the algebraic expression is called a polynomial.

Some polynomials are classified by the number of terms they contain. A monomial is a
polynomial with one term, a binomial is a polynomial with two terms, and a trinomial is
a polynomial with three terms. Polynomials with four or more terms have no special
names. Thus, they are simply called polynomials.

The degree of a term in a polynomial is the sum of the exponents of its variable factors.
The degree of a constant term is zero.

The degree of a polynomial is the largest degree of its terms. The term which determines
the degree of the polynomial is called the leading term of the polynomial and its
coefficient is called the leading coefficient.

The conventional method of writing a polynomial is to write the terms in order of
decreasing degree. When a polynomial is written in this manner, it is said to be in
descending order.

Simplifying Polynomials
Two terms are called like terms if:
1. They are both constant terms, or
2. Both variable terms contain the same variables with the same exponents on each
variable.

We collect like terms in an algebraic expression in order to simplify it. This means that
we add the coefficients of the like terms, keeping the variable parts unchanged.

Evaluating Polynomials
A polynomial, like any algebraic expression, represents different values depending on the
value(s) of its variable(s). To evaluate a polynomial, we substitute the value (s) of its
variable(s) into the expression, and then determine the value of the resulting numeric
expression.

Section 4.3 & Section 4.6 (Exercises 23-34) Addition and Subtraction of Polynomials

Addition of Polynomials
To add two polynomials enclosed in a set of parentheses :
1. Remove the parentheses, and then
2. Combine like terms

When we add two polynomials a horizontal or vertical format can be used.

To use the vertical format:
1. Rewrite each polynomial in descending order,
2. Arrange the polynomials with like terms in the same column, leaving spaces for
any missing terms
3. Then combine the terms in each column.

The Opposite of a Polynomial
Recall that the opposite of a sum property of real numbers states the following fact:
− (a + b) = −a + (− b). What this means is that if we want the opposite of an expression,
we change the sign of every term in that expression. This is the same as multiplying
every term by – 1.

Subtraction of Polynomials
Recall the definition for subtraction of real numbers: a − b = a + (− b). So, since
polynomials have the same properties of real numbers, we can utilize this definition for
the subtraction of polynomials.

To subtract polynomials enclosed in a set of parentheses:
1. Rewrite the subtraction problem as an addition problem using the definition
a − b = a + (− b).
2. Perform the addition.

Applications
Polynomials describe many situations in the real world. If we can find the polynomial
that describes a particular situation, we can analyze the situation more closely and then
make appropriate decisions regarding the situation.

Section 4.4 & Section 4.6 (Exercises 35-95) Multiplication of Polynomials

To multiply a monomial by a monomial
1. Use the commutative and associative properties to rearrange factors (constants
next to each other and variables next to each other)
2. Multiply the coefficients
3. Simplify the variable factors using the rules of exponents .

To multiply a polynomial by a monomial
1. Use the distributive law to multiply every term in the polynomial by the
monomial factor.
2. Simplify each term
3. Collect like terms

To multiply a polynomial by a polynomial
1. Use the distributive law to distribute every term of one factor to every term of the
second polynomial factor.
2. Simplify each term
3. Collect like terms

Section 4.5 Special Products of Binomials

To multiply any two binomial factors
1. Use the distributive law to distribute each term of the first binomial factor to the
second binomial factor as follows.

(a + b)(c + d ) = a(c + d )+ b(c + d )

2. Use the distributive law again as follows.


3. Simplify each term.

4. Collect like terms

Sometimes, this process is called the FOIL method, where F stands for the first terms, O
stands for the outside terms, I for the inner terms, and L for the last terms. Graphically,
this looks like:

Special Binomial Products
Product of a Sum and Difference: (a.k.a. A Difference of Squares)

(A+ B)(A− B) = A2 − B2

Binomial Squares: (a.k.a. Perfect Square Trinomials )

Section 4.7 Division of Polynomials

To divide a polynomial by a monomial
1. Use the distributive property of division, which is
2. Simplify each term
3. Collect like terms

To divide a polynomial by a binomial
We can perform division of a polynomial by a binomial provided that the degree of the
polynomial is greater than or equal to the degree of the binomial. This requires a process
called polynomial long division, which is ana logous to the long division of arithmetic.

To perform polynomial long division of the expression a ÷b
1. Write the numerator and denominator in descending order.
2. Fill in any “missing terms”, using 0 for the coefficients.
3. Write the division problem into a long division format .
4. Perform the division by determining what you must multiply the leading term of
the binomial by to obtain the leading term of the polynomial.
5. Continue the division operation until
a. The remainder is 0, or
b. The degree of the remainder is less than the degree of the divisor.
6. Check your answer as follows: polynomial = binomial · quotient + remainder

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