Transcript Chapter 9 Section 1

Chapter 9 Section 1
Polynomials
Vocabulary
Polynomial: The sum of one or more
monomials is called a polynomial.
Monomial: A monomial is a number, a
variable, or a product of numbers and
variables that have only positive
exponents.
Binomial: A polynomial with two terms is a
binomial.
Trinomial: A polynomial with three terms is
a trinomial.
What You’ll Learn
You’ll learn to identify and classify
polynomials and find their degree.
Why it is important
One example of why it is important is
medicine. Doctors can use polynomials
to study the heart.
A cube is a solid figure in which all the
faces are square. Suppose you wanted
to paint the cube shown below. You
would need to find the surface area of
the cube to determine how much paint
to buy.
x ft.
x ft.
x ft.
The area of each face of the cube is x ∙ x
or x2. There are six faces to paint.
x2 + x2 + x2 + x2 + x2 + x2 = 6x2
So, the surface area of the cube is 6x2
square feet.
The expression 6x2 is called a monomial.
A monomial is a number, a variable, or a
product of numbers and variables that
have only positive exponents. A
monomial cannot have a variable as an
exponent.
Monomials
Not Monomials
-4
A number
2x
Has a variable as an
exponent
Y
A variable
x2 + 3
Includes addition
a2
The product of variables 5a-2
Includes a negative
exponent
The product of numbers
and variables
Includes division
½ x2y
3
x
Example 1
Determine whether each expression is a
monomial. Explain why or why not.
-6ab
-6ab is a monomial. It is the product
of a number and variables.
Example 2
Determine whether each expression is a
monomial. Explain why or why not.
m2 - 4
m2 – 4 is not a monomial, because it
includes subtraction.
Your Turn
Determine whether each expression is a
monomial. Explain why or why not.
10
10 is a monomial, because it is a
number.
Your Turn
Determine whether each expression is a
monomial. Explain why or why not.
5z-3
5z-3 is not a monomial, because it
includes a negative exponent.
Your Turn
Determine whether each expression is a
monomial. Explain why or why not.
6
x
This is not a monomial, because it
includes division.
Your Turn
Determine whether each expression is a
monomial. Explain why or why not.
x2
x2 is a monomial, because it is a
product of variables.
The sum of one or more monomials is
called a polynomial. For example,
x3 + x2 + x + 2
is a polynomial. The terms of the
polynomial are x3, x2, x, and 2.
Special names are given to polynomials
with two or three terms. A
polynomial with two terms is a
binomial. A polynomial with three
terms is a trinomial. Here are some
examples.
Binomial
Trinomial
x+2
a+b+c
5c - 4
x2 + 5x - 7
4w2 - w
3a2 + 5ab + 2b2
Example 3
State whether each expression is a
polynomial. If it is a polynomial,
identify it as a monomial, binomial, or
trinomial.
2m – 7
The expression 2m – 7 can be written
as 2m + (-7). So, it is a polynomial.
Since it can be written as the sum of
two monomials, 2m and -7, it is a
binomial.
Example 4
State whether each expression is a
polynomial. If it is a polynomial,
identify it as a monomial, binomial, or
trinomial.
x2 + 3x – 4 - 5
The expression x2 + 3x – 4 – 5 can be
written as x2 + 3x + (-9).
So, it is a polynomial. Since it can be
written as the sum of three
monomials, it is a trinomial.
Example 5
State whether each expression is a
polynomial. If it is a polynomial,
identify it as a monomial, binomial, or
trinomial.
5 - 3
2x
The expression is not a polynomial since
it is not a monomial. It contains
division.
Your Turn
State whether each expression is a
polynomial. If it is a polynomial,
identify it as a monomial, binomial, or
trinomial.
5a – 9 + 3
Yes, Binomial
Your Turn
State whether each expression is a
polynomial. If it is a polynomial,
identify it as a monomial, binomial, or
trinomial.
4m-2 + 2
No, Cannot have a negative exponent
Your Turn
State whether each expression is a
polynomial. If it is a polynomial,
identify it as a monomial, binomial, or
trinomial.
3y2 – 6 + 7y
Yes, Trinomial
The terms of a polynomial are usually
arranged so that the powers of one
variable are in descending or
ascending order.
Polynomial
Descending Order
Ascending Order
2x + x2 + 1
x2 + 2x + 1
1 + 2x + x2
3y2 + 5y3 + y
5y3 + 3y2 + y
y + 3y2 + 5y3
x2 + y2 + 3xy
x2 + 3xy + y2
y2 + 3xy + x2
2xy + y2 + x2
y2 + 2xy + x2
x2 + 2xy + y2
Degree
The degree of a monomial is the sum
of the exponents of the variables.
Monomial
Degree
-3x2
2
5pq2
1 + 2 = 3
2
0
To find the degree of a polynomial,
you must find the degree of each
term. The degree of the polynomial
is the greatest of the degrees of its
term.
Polynomial
Terms
Degree of the
Terms
Degree of the
Polynomial
2n + 7
2n, 7
1, 0
1
3x2 + 5x
3x2, 5x
2, 1
2
a6 + 2a3 + 1
a6, 2a3, 1
6, 3, 0
6
5x4 – 4a2b6 + 3x
5x4, 4a2b6, 3x
4, 8, 1
8
Example 6
Find the degree of each polynomial.
5a2 + 3
Term
Degree
5a2
2
3
0
So, the degree of 5a2 + 3 is 2.
Example 6
Find the degree of each polynomial.
6x2 – 4x2y – 3xy
Term
Degree
6x2
2
4x2y
2 + 1 or 3
3xy
1 + 1 or 2
So, the degree of 6x2 – 4x2y – 3xy
is 3.
Your Turn
Find the degree of each polynomial.
3x2 – 7x
2
Your Turn
Find the degree of each polynomial.
8m3 – 2m2n2 + 5
4