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6-3 Polynomials
Holt McDougal Algebra 1
6-3 Polynomials
A monomial is a number, a variable, or a product
of numbers and variables with whole-number
exponents.
Holt McDougal Algebra 1
6-3 Polynomials
Monomials
NOT a monomial
5+z
2/n
4a
x-1
Holt McDougal Algebra 1
Reason
A sum is not a monomial
A monomial cannot
have a variable
denominator
A monomial cannot
have a variable
exponent
The variable must have
a whole number
exponent.
6-3 Polynomials
Monomial
10
3x
Degree
0
1
1+2=3
-1.8m5
5
The degree of a monomial is the sum of the
exponents of the variables. A constant has
degree 0.
Holt McDougal Algebra 1
6-3 Polynomials
Example 1: Finding the Degree of a Monomial
Find the degree of each monomial.
A. 4p4q3
The degree is 7.
B. 7ed
The degree is 2.
C. 3
The degree is 0.
Holt McDougal Algebra 1
6-3 Polynomials
Check It Out! Example 1
Find the degree of each monomial.
a. 1.5k2m
The degree is 3.
b. 4x
The degree is 1.
c. 2c3
The degree is 3.
Holt McDougal Algebra 1
6-3 Polynomials
A polynomial is a monomial or a sum or
difference of monomials.
Each monomial in a polynomial is called a
term.
The degree of a polynomial is the
degree of the term with the greatest
degree.
Holt McDougal Algebra 1
6-3 Polynomials
Polynomials
Degree of
polynomial
2 x  x  5x  12
3
Leading
Coefficient
Holt McDougal Algebra 1
2
Constant
term
6-3 Polynomials
Special Polynomials
• Binomial
– Polynomial with two terms
• Trinomial
– Polynomial with three terms
Holt McDougal Algebra 1
6-3 Polynomials
Example 2: Finding the Degree of a Polynomial
Find the degree of each polynomial.
A. 11x7 + 3x3
The degree of the polynomial is
the greatest degree, 7.
B.
The degree of the polynomial is the greatest degree, 4.
Holt McDougal Algebra 1
6-3 Polynomials
Check It Out! Example 2
Find the degree of each polynomial.
a. 5x – 6
The degree of the polynomial
is the greatest degree, 1.
b. x3y2 + x2y3 – x4 + 2
The degree of the polynomial is
the greatest degree, 5.
Holt McDougal Algebra 1
6-3 Polynomials
The terms of a polynomial may be written in
any order. However, polynomials that
contain only one variable are usually written
in standard form.
The standard form of a polynomial that
contains one variable is written with the
terms in order from greatest degree to
least degree. When written in standard
form, the coefficient of the first term is
called the leading coefficient.
Holt McDougal Algebra 1
6-3 Polynomials
Example 3A: Writing Polynomials in Standard Form
Write the polynomial in standard form. Then
give the leading coefficient.
6x – 7x5 + 4x2 + 9
The standard form is –7x5 + 4x2 + 6x + 9. The leading
coefficient is –7.
Holt McDougal Algebra 1
6-3 Polynomials
Example 3B: Writing Polynomials in Standard Form
Write the polynomial in standard form. Then
give the leading coefficient.
y2 + y6 – 3y
The standard form is y6 + y2 – 3y. The leading
coefficient is 1.
Holt McDougal Algebra 1
6-3 Polynomials
Remember!
A variable written without a coefficient has a
coefficient of 1. Remember “the understood 1”
y5 = 1y5
Holt McDougal Algebra 1
6-3 Polynomials
Check It Out! Example 3a
Write the polynomial in standard form. Then
give the leading coefficient.
16 – 4x2 + x5 + 9x3
The standard form is x5 + 9x3 – 4x2 + 16. The leading
coefficient is 1.
Holt McDougal Algebra 1
6-3 Polynomials
Check It Out! Example 3b
Write the polynomial in standard form. Then
give the leading coefficient.
18y5 – 3y8 + 14y
The standard form is –3y8 + 18y5 + 14y. The leading
coefficient is –3.
Holt McDougal Algebra 1
6-3 Polynomials
Some polynomials have special names based on
their degree and the number of terms they have.
Degree
Name
Terms
Name
0
Constant
1
Monomial
1
Linear
2
Binomial
2
Quadratic
Trinomial
3
4
Cubic
Quartic
3
4 or
more
5
Quintic
6 or more
Holt McDougal Algebra 1
6th,7th,degree
and so on
Polynomial
6-3 Polynomials
Example 4: Classifying Polynomials
Classify each polynomial according to its
degree and number of terms.
A. 5n3 + 4n
cubic binomial.
B. 4y6 – 5y3 + 2y – 9
6th-degree polynomial.
C. –2x
linear monomial.
Holt McDougal Algebra 1
6-3 Polynomials
Check It Out! Example 4
Classify each polynomial according to its
degree and number of terms.
a. x3 + x2 – x + 2
cubic polynomial.
b. 6
constant monomial.
c. –3y8 + 18y5 + 14y
8th-degree trinomial.
Holt McDougal Algebra 1
Example 2
6-3 Polynomials
Tell whether is a polynomial. If it is a polynomial, find
its degree and classify it by the number of its terms.
Otherwise, tell why it is not a polynomial.
Expression Is it a polynomial?
Classify by degree and
number of terms
a.
9
Yes
constant monomial
b.
c.
d.
e.
2x2 + x – 5
Yes
Quadratic trinomial
6n4 – 8n
No; variable exponent
n– 2 – 3
No; negative exponent
7bc3 + 4b4c
Yes
Holt McDougal Algebra 1
Quintic binomial
6-3 Polynomials
Example 5: Application Continued
A tourist accidentally drops her lip balm off the
Golden Gate Bridge. The bridge is 220 feet from
the water of the bay. The height of the lip balm
is given by the polynomial –16t2 + 220, where t
is time in seconds. How far above the water will
the lip balm be after 3 seconds?
After 3 seconds the lip balm will be 76 feet
from the water.
Holt McDougal Algebra 1
6-3 Polynomials
Check It Out! Example 5
What if…? Another firework with a 5-second
fuse is launched from the same platform at a
speed of 400 feet per second. Its height is
given by –16t2 +400t + 6. How high will this
firework be when it explodes?
1606 feet
Holt McDougal Algebra 1
6-3 Polynomials
Check It Out! Example 4a
Simplify. All variables represent nonnegative
numbers.
= xy
Holt McDougal Algebra 1
6-3 Polynomials
Solve for the missing exponent.
5d 
?
4
 625d
5a  10a
6
Holt McDougal Algebra 1

5 ?
16
 5000a
21