§6.1
Adding and Subtracting Polynomials
Polynomials
A polynomial is a single term or the sum of two
or more terms
containing variables with whole number exponents.
Consider the polynomial:
This polynomial contains four terms. It is customary to
write the terms in order
of descending powers of the variable. This is the standard form of a
polynomial.
Here are two other polynomials which are written in standard form.
The degree of a polynomial is the greatest degree
of any term of the
polynomial. The degree of a term axnym is (n +m)
and the coefficient of the term is a. If there is
exactly one
term of greatest degree, it is called the leading term. It’ s
coefficient is called the leading coefficient. Consider the
polynomial:
3 is the leading coefficient. The degree is 4.
The Degree of axn
• If a ≠ 0, the degree of axn is n. The degree of a nonzero constant
term is 0. The constant 0 has no defined degree.
Degree of a Polynomial
• The degree of a polynomial is the degree of its
highest order
term.
• Example:
Degree 3 Polynomial:
Degree 4 Polynomial:
Special Polynomials
• Monomial: A polynomial with one term .
• Binomial: A polynomial with two terms.
• Trinomial: A polynomial with three terms.
Example:
This is a 4th degree trinomial.
Polynomials
| The Degree of axn |
If , the degree of is axn. The degree of a
nonzero constant is 0. The constant 0 has no defined
degree. |
| |
| Adding Polynomials |
Polynomials are added by removing the parentheses
that
surround each polynomial (if any) and then combining
like terms. |
| |
| Subtracting Polynomials |
To subtract two polynomials, change the sign of
every
term of the second polynomial. Add this result to the
first polynomial. |
Adding Polynomials
• Polynomials are added by combining like terms.
• Like terms are terms containing exactly the same
variables to the same powers.
• Example:
EXAMPLE
Adding Polynomials
Add :
Group like terms.
Combine like terms.
EXAMPLE
Add:
SOLUTION
|
|
 |
Remove parentheses |
 |
Rearrange terms so that
like terms are adjacent |
 |
Combine like terms |
Subtracting Polynomials
EXAMPLE
Subtract :
 |
|
 |
Add the opposite of the
polynomial being subtracted. |
 |
Group like terms. |
 |
Combine like terms. |
EXAMPLE
Subtract
SOLUTION
 |
|
 |
Change subtraction to
addition and change the
sign of every term of the
polynomial in
parentheses. |
 |
Rearrange terms |
 |
Combine like terms |
Polynomial Functions
is an example of a polynomial function. In a
polynomial
function, the expression that defines the function is a
polynomial.
How do you evaluate a polynomial function? Use
Substitution.
Graphs of Polynomial Functions
Polynomial functions of degree 2 or higher have
graphs
that are smooth and continuous.
By smooth, we mean that the graph contains only
rounded corners with no sharp corners.
By continuous, we mean that the graph has no breaks
and can be drawn without lifting the pencil from the page.
Graphs of Polynomials
EXAMPLE
The graph below does not represent
a polynomial function. Although it
has a couple of smooth, rounded
corners, it also has a sharp corner
and a break in the graph. Either one
of these last two features disqualifies
it from being a polynomial function.
§6.2
Multiplying Polynomials
Product Rule
When multiplying exponential expressions with
the same base, add the exponents.
A time to add…
EXAMPLE
Multiply x3·x5
We used the Product Rule. When multiplying
exponential ex pressions with the same base,
add the exponents.
Note that the product rule does not apply to exponential
expressions with different bases.
EXAMPLE
Multiply 23·22
We used the Product Rule. When multiplying
exponential expressions with the same base, add the
exponents. Note: We do not change the base!
Power Rule for Exponents
The Power Rule
When an exponential expression is raised to a power,
multiply the exponents.
A time to multiply…
EXAMPLE
Find (x2)5
We used the Power Rule. When an exponential
expression is raised to a power, multiply the
exponents.
Power Rule
EXAMPLE Find (34)2
We used the Power Rule. When an exponential
expression is raised to a power, multiply the
exponents. Note: We do not change the base!
Products to Powers Rule for Exponents
When a product is raised to a power, raise each
factor to the power.
EXAMPLE
Find (5x3)2
 |
Products to a Power Rule |
 |
Power Rule |
 |
|
First, we used the Products to a Power Rule. When a
product is raised to a
power, raise each factor to the power.
Second, we used the Power Rule. When an exponential expression is raised
to a power, multiply the exponents.
Multiplying Monomials
To multiply monomials with the
same variable base, multiply the
coefficients and then multiply the
variable parts. Use the product rule
for exponents to multiply the
variables: Keep the variable and add
the exponents.
EXAMPLE Multiply (3x2)(5x7)
 |
Multiply the coefficients and
multiply the variables. |
 |
Add the exponents. |
 |
Simplify. |
Multiplying a Monomial and a Polynomial
To multiply a monomial and a polynomial that
is not a monomial, use the
distributive property to multiply each
term of the polynomial by the monomial
EXAMPLE
Multiply 3x(2x2 - 5x + 7)
 |
|
 |
Use the Distributive
Property. |
 |
Multiply the coefficients and
add exponents. |
 |
Simplify. |
Multiplying Polynomials
Multiply each term of one polynomial by each
term of the other polynomial. Then combine
like terms.
For example, if multiplying a binomial by a
trinomial –you would have 6 products
initially before you combined like terms.
Multiply (3x + 2)(2x –7)
 |
Multiply the second binomial by each term of the
first binomial. |
 |
Use the distributive property. |
 |
Multiply. Note the 4 products here. That’s
because 2 times 2 = 4. |
 |
Simplify. |
EXAMPLE
Multiply
SOLUTION
 |
|
 |
Rearrange factors |
 |
Multiply coefficients and
add exponents |
 |
Simplify |
EXAMPLE
Multiply 
 |
|
 |
Distribute |
 |
Multiply coefficients and
add exponents |
EXAMPLE
Multiply 
Note this: We multiply each term of the
binomial by each term of the trinomial.
We get 6 products in all.
 |
|
 |
Multiply the trinomial by
each term of the binomial |
 |
Distribute |
 |
Multiply coefficients and
add exponents |
 |
Simplify |
§6.3
Special Products
Special Products
In this section we will use the distributive property
to develop patterns
that can help you in multiplying some special binomials quickly.
We will find the product of two binomials using a method called FOIL. You
will be thinking…”first two, outer two, inner two, last two” before the
section is over.
We will learn a formula for finding the square of a binomial sum. You will
learn a formula for finding the product of the sum and difference of two
terms.
And whether you choose to take a handy shortcut and use these
formulas or simply use polynomial multiplication will be left to you to
decide.
Multiplying Polynomials - FOIL
The FOIL Method
Product of Two Binomials
Distribute each term in the first binomial
through each term of the second binomial.
Multiplying Polynomials - FOIL
EXAMPLE
Multiply
SOLUTION
Multiply
Combine like terms
FOIL Method
EXAMPLE
Multiply (5x + 2)(x + 7)
EXAMPLE
Multiply (3x -5)(2x -8)
First two, Outer two, Inner two, Last two…
The Square of a Binomial Sum
(A + B)2= A2+ 2AB+ B2
The square of a binomial sum is the first
term squared
plus two times the product
of the terms plus the last term squared.
Multiplying the Sum and Difference of Two Terms
(A + B)(A –B) = A2 -B2
The product of the sum and the difference
of the same two
terms is the square of the first
term minus the square of the second term.
EXAMPLE
Multiply (x –5)(x + 5)
Since this is the product of a sum and a
difference, we
use the rule:
The Square of a Binomial Sum
EXAMPLE
Find (x + 7 )2
Since this is the square of a binomial
sum, we use the
rule:
The Square of a Binomial Difference
(A -B)2= A2 -2AB+ B2
The square of a binomial difference is the
first term
squared minus two times the product of the terms
plus the last term squared.
EXAMPLE
Find (3x -5 )2
Since this is the square of a binomial
difference, we use
the rule:
The Square of a Binomial Sum
EXAMPLE
Find (x + 7 )2
Since this is the square of a binomial
sum, we use the
rule:
The Square of a Binomial Difference
EXAMPLE
Find (3x -5 )2
Since this is the square of a binomial
difference, we use
the rule:
Multiplying Polynomials – Special Formulas
| The Square of a Binomial Sum |
 |
| |
| The Square of a Binomial Difference |
 |
| |
The Product of the Sum and
Difference of Two Terms |
 |
EXAMPLE
Multiply 
SOLUTION
Use the special-product formula shown.
| |
 |
+ |
2 Product
of the Terms |
+ |
 |
= Product |
 |
 |
+ |
2 4x y |
+ |
y2 |
 |
EXAMPLE
Multiply 
SOLUTION
Use the special-product formula shown.
| |
|
- |
2 Product
of the Terms |
+ |
|
= Product |
 |
 |
- |
2 3x 4y |
+ |
 |
 |
EXAMPLE
Multiply 
SOLUTION
Use the special-product formula shown.
§6.4
Polynomials in Several Variables
Polynomials in Several Variables
A polynomial containing two or more variables
is called a polynomial in several variables. An
example of a polynomial in two variables is:
Evaluating a Polynomial in Several Variables
1. Substitute the given value for each variable .
2.Perform the resulting computation using the order of operations .
EXAMPLE
Evaluate 
for
x = 3 and y = -1.
1. Substitute the given value for each variable.
2. Perform the resulting computation using the order of
operations.
Adding and Subtracting Polynomials in Several Variables
•Polynomials in several variables are added
by combining
like terms.
•Polynomials in several variables are subtracted
by adding the first polynomial
and the opposite of the second polynomial.
Like terms are terms containing exactly the same variables to the
same powers.
Adding Polynomials with Two Variables
EXAMPLE
Add :
 |
|
 |
Group like terms. |
 |
Combine like terms. |
Multiplying Polynomials in Several Variables
EXAMPLE
Multiply coefficients and add exponents on variables
with the same base.
 |
|
 |
Regroup. |
 |
Multiply the coefficients
and add the exponents. |
EXAMPLE
Multiply each term of the polynomial by the monomial
 |
|
 |
Use the distributive
property. |
 |
Multiply the coefficients
and add the exponents. |
EXAMPLE
Multiply each term of one polynomial by each term in the
other
polynomial. (For Binomial · Binomial use FOIL.)
 |
|
 |
Use the distributive
property. |
 |
Multiply the coefficients
and add the exponents. |
EXAMPLE
Determine the coefficient of each term, the degree of each
term, the degree of the polynomial, the leading term, and the
leading coefficient of the polynomial.
SOLUTION
| Term |
Coefficient |
Degree (Sum of Exponents on
the Variables) |
| 12x4y |
12 |
4 + 1 = 5 |
| -5x3y7 |
-5 |
3 + 7 = 10 |
| -x2 |
-1 |
2 + 0 = 2 |
| 4 |
4 |
0 + 0 = 0 |
Polynomials
CONTINUED
The degree of the polynomial is the greatest degree of all
its
terms, which is 10. The leading term is the term of the
greatest degree, which is -5x3y7 . Its coefficient, -5, is the
leading coefficient.
Subtracting Polynomials
EXAMPLE
Subtract 
SOLUTION
 |
|
 |
Change subtraction to
addition and change the
sign of every term of the
polynomial in
parentheses. |
 |
Rearrange terms |
 |
Combine like terms |
§6.5
Dividing Polynomials
The Quotient Rule
When dividing exponential expressions with the same
nonzero
base, subtract the exponent in the denominator from the
exponent in the numerator. Use this difference as the
exponent on the common base .
The Quotient Rule for Exponents
EXAMPLE
Divide:
EXAMPLE
Divide:
=53 or 125
EXAMPLE
Divide :
But we know any nonzero
expression divided by itself is 1.
So
The Zero Exponent Rule
If b is any real number other than 0,
Zero as an Exponent
EXAMPLES
 |
5 is raised to the 0 power. |
 |
2xy is raised to the 0 power. |
 |
Only y is raised to the 0 power |
 |
Only 2 is raised to the 0 power. |
Quotients to Powers Rule for Exponents
If a and b are real numbers and b is nonzero, then
When a quotient is raised to a power, raise the numerator
to
the power and divide by the denominator raised to the power.
Quotients-to-Powers Rule for Exponents
EXAMPLES
Quotients to Powers Rule
EXAMPLE
 |
|
 |
Cube the numerator
and denominator. |
 |
Cube each factor in the
numerator. |
 |
Simplify. |
Dividing Monomials
To divide monomials, divide the coefficients
and then
divide the variables (by subtracting
exponents). Use the quotient rule for
exponents to divide the variable factors. Keep
the variable and subtract the
exponents.
Division of Polynomials by Monomials
Now we will look at dividing a polynomial by a
monomial.
Division of a polynomial by a monomial is relatively easy –
you just divide each
term of the polynomial by the
monomial. The number of separate divisions you will
have is the number of terms in the polynomial.
Dividing Monomials
EXAMPLE
 |
|
 |
Divide the coefficients, 5/10 = 1/2, then
divide the variables by subtracting
exponents. |
 |
Simplify. |
EXAMPLE
Divide:
 |
|
 |
Divide the coefficients, 6/2 = 3, then divide
the variables by subtracting exponents. |
 |
Simplify. |
Division of Polynomials
Dividing a Polynomial by a
Monomial |
To divide a polynomial by a monomial,
divide each
term of the polynomial by
the monomial. |
EXAMPLE
Divide:
SOLUTION
 |
Express the division in a vertical
format. |
 |
Divide each term of the polynomial
by the monomial. Note the 3
separate quotients. |
 |
Simplify each quotient. |
Dividing a Polynomial by a Monomial
Divide :
 |
|
 |
Divide each term of the polynomial by
the monomial. |
 |
Divide the coefficients, then divide the
variables by subtracting exponents. |
 |
Simplify. |
§6.6
Dividing Polynomials by Binomials
Division of Polynomials
In the last section we looked at dividing by a monomial.
In
this section we will look at dividing by a binomial.
Division of a polynomial by a monomial was a relatively easy task as
we saw –we
just divided each term of the polynomial by the
monomial. The number of separate divisions we had was the
number of terms in the polynomial.
The second case, that of dividing a polynomial by a binomial or any
other
polynomial having more than one term, is more difficult. This
requires a process
of long division.
We will now consider the harder problem –that of
dividing a polynomial by a binomial.
The four steps that you remember using in long division of
whole numbers
–divide, multiply, subtract, bring down
the next term –form the same repetitive
procedure for
polynomial long division.
Carefully consider and try to remember the four terms
illustrated on the next
slide. These terms are: quotient,
divisor, dividend, and remainder.
EXAMPLE
EXAMPLE
Divide:
SOLUTION
 |
Arrange the terms of the
dividend, , and 
the divisor, (x + 2), in descending
powers of x. |
 |
Divide x3 (the first term in the
dividend) by x (the first term in
the divisor). Align like terms. |
CONTINUED
 |
Multiply each term in the divisor
(x + 2) by x2, aligning terms of
the product under like terms in
the dividend. |
 |
Subtract x3 +2x2 from x3 +5x2
by changing the sign of each
term in the lower expression and
then adding. |
CONTINUED
 |
Bring down 7x from the original
dividend and add algebraically to
form a new dividend. |
 |
Find the second term of the
quotient. Divide the first term
of 3x2 +7x by x, the first term
of the divisor. |
CONTINUED
Multiply the divisor (x + 2) by
3x, aligning under like terms in
the new dividend. Then subtract.
CONTINUED
Bring down 2 from the original
dividend and add algebraically to
form a new dividend.
Find the third term of the
quotient, 1. Divide the first term
of x + 2 by x, the first term of the
divisor.
Multiply the divisor by 1,
aligning under like terms in the
new dividend. Then subtract to
obtain the remainder of 0.
CONTINUED
The quotient is 
and the
remainder is 0. We will not list
a remainder of 0 in the answer. Thus
Long Division of Polynomials
| Long Division of Polynomials |
| 1) Arrange the terms of both the dividend and the
divisor in descending powers of any variable. |
| 2) Divide the first term in the dividend by the
first term in the divisor. The result is the first term of the quotient |
| 3) Multiply every term in the divisor by the first
term in the quotient. Write the resulting product beneath the dividend
with like terms lined up . |
| 4) Subtract the product from the dividend. |
| 5) Bring down the next term in the original
dividend and write it next to the remainder to form a new dividend. |
| 6) Use this new expression as the dividend and
repeat this process until the remainder can no longer be divided. This
will occur when the degree of the remainder (the highest exponent on a
variable in the remainder) is less than the degree of the divisor. |
EXAMPLE
Divide:
SOLUTION
We write the dividend, ,
as
to keep all like terms aligned. For the
same reason, we write the divisor, 
as
Note that when terms are missing in the dividend, you
should insert the term
using a coefficient of 0. This is to keep the terms aligned. The term with the 0
coefficient is still equal to 0, but that term serves as an effective
placeholder.
CONTINUED
CONTINUED
The division process is finished because the degree of -2x, which
is 1, is less than the degree of the divisor , which is
. The
answer is
Important to Remember:
To divide by a polynomial containing more than one term, use long
division. If
necessary, arrange the dividend in descending powers
of the variable. Do the
same with the divisor. If a power of a
variable is missing in the dividend, add
that term using a
coefficient of 0.
Repeat the four steps of the long-division process –divide,
multiply, subtract,
bring down the next term –until the degree of
the remainder is less than the
degree of the divisor.
When the degree of the remainder is less than the degree of the
divisor –you
know you are done!
§6.7
Negative Exponents and Scientific Notation
Negative Exponents and Scientific Notation
We frequently encounter very large or very small numbers.
Think about the size of the national debt (BIG!) or the
diameter of an atom
(small!). In this section we use
exponents to put really big or really small
numbers into
perspective.
We will first define negative exponents and then will use
these for writing
numbers in scientific notation.
We begin by reviewing our exponent rules.
Properties of Exponents
| Exponent Rules |
|
| Product Rule |
 |
When multiplying exponential expressions with the
same base, add the exponents. |
| Quotient Rule |
 |
When dividing exponential expressions with
the same nonzero base, subtract the exponent
in the denominator from the exponent in the
numerator. |
| Exponent Rules |
Examples |
| Product Rule |
 |
 |
| Quotient Rule |
 |
 |
Properties of Exponents
The Zero Exponent Rule:
If b is any real number other than 0, then 
Negative Exponent Rule: If b is any real number
other than 0 and n is a natural number, then
and
Negative Exponents
Write x-4 with positive exponents .
Write 5-3 with positive exponents.
Negative Exponents in Numerators and Denominators
If b is any real number other than 0 and n is a natural
number,
then
When a negative number appears as an exponent, switch the
position of the base (from numerator to denominator or
denominator to numerator) and make the exponent positive.
The sign of the base does not change.
EXAMPLE
Write with positive exponents.
SOLUTION
When a negative number appears as
an exponent, switch the position of
the base. Here, the y-3 moves from
numerator to denominator as y3 and
the x-2 moves from the denominator
to numerator as x2 The sign of the
base does not change.
Properties of Exponents
| Exponent Rules |
Examples |
| Zero Exponent Rule |
 |
| Negative Exponent Rule |
 |
| Exponent Rules |
Examples |
Negative Exponents in
Numerators and
Denominators |
 |
| Power Rule |
 |
Exponent
Rules |
Examples |
Products to
Powers |
 |
 |
Quotients
to Powers |
 |
 |
Simplifying Exponential Expressions
Simplification
Techniques |
Examples |
If necessary, remove
parentheses by using the
Products to Powers Rule or the
Quotient to Powers Rule. |
 |
If necessary, simplify powers to
powers by using the Power
Rule. |
 |
Simplification
Techniques |
Examples |
Be sure each base appears only
once in the final form by using
the Product Rule or Quotient
Rule |
 |
If necessary, rewrite
exponential expressions with
zero powers as 1. Furthermore,
write the answer with positive
exponents by using the
Negative Exponent Rule |
 |
Of importance to note…
An exponential expression is “simplified” when
Each base occurs only once.
No parentheses appear.
No powers are raised to powers.
No negative or zero exponents appear.
| EXAMPLE |
Simplify |
 |
Cube each factor in the numerator. |
 |
Multiply powers using (bm)n = bmn. |
 |
Division with the same base, subtract exponents. |
 |
When a negative number appears as an exponent,
switch
the position of the base. The x-2 moves from numerator to
denominator as x2. |
Properties of Exponents
Of importance to note…
Be aware that a negative exponent does not make the
value
of the expression negative. The sign of the
exponent in no way affects the sign
of the term.
The negative exponent, if it could talk, would
simply be saying:
“Take the reciprocal.”
Scientific Notation
At times you may find it necessary to work with really
large
numbers, or alternately, really small numbers. In this section, you
will
learn how to write these often cumbersome numbers in
scientific notation.
A number is written in scientific notation when it is expressed as the
product
of a number between one and ten and some power of ten.
Study Tip:
“Bignonnegative numbers” have positive powers of ten when
written in scientific notation. That is, if the absolute value of
a number is
BIG (greater than 10), it will have a positive exponent in
scientific notation.
Small nonnegative numbers have negative powers of ten when
written in scientific
notation. That is, if the absolute value of
a number is small (less than 1), it
will have a negative exponent in
scientific notation.
Converting from Decimal to Scientific Notation
•Determine a, the numerical factor. Move the decimal point in
the given number
to obtain a number greater than or equal to
1 and less than 10.
•Determine n, the exponent on 10n. The absolute value of n is
the number of
places the decimal was moved. The exponent
n is positive if the given number is
great than 10 and negative
if the given number is between 0 and 1.
Converting from Decimal to Scientific Notation
(Write the number in the form a* 10n) |
1) Determine a, the numerical factor. Move the
decimal point in the
given number to obtain a number whose absolute value is between 1 and
10, including 1. |
2) Determine n, the exponent on
10n. The absolute value of n is the
number of places the decimal point was moved. The exponent n is
positive if the decimal point was moved to the left, negative is the
decimal point was moved to the right, and 0 if the decimal point was not
moved. |
Scientific Notation to Decimal Notation
EXAMPLE
Write each number in decimal notation.
SOLUTION
EXAMPLE
Write each number in scientific notation.
324,510,000,000,000,000
0.0000000859
-4395
SOLUTION
Scientific Notation
EXAMPLE
Perform the indicated computation, writing the answer in
scientific notation.
SOLUTION
 |
|
 |
Regroup factors |
 |
Multiply |
 |
Simplify |
 |
Rewrite in scientific |
 |
|
|
