A monomial is a number, a variable, or a product of numbers and variables with whole-number exponents. A monomial may be a constant or a single variable.

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A monomial is a number, a variable, or a product
of numbers and variables with whole-number
exponents. A monomial may be a constant or a
single variable.
The degree of a monomial is the sum of the
exponents of the variables. A constant has degree
0.
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Additional Example 1 Finding the Degree of a
Monomial
Find the degree of each monomial.
Add the exponents of the variables 4 3 7.
The degree is 7.
B. 7ed
A variable written without an exponent has an
exponent of 1. 1 1 2.
The degree is 2.
C. 3
There is no variable, but you can write 3 as 3x0.
The degree is 0.
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Check It Out! Example 1
Find the degree of each monomial.
Add the exponents of the variables 2 1 3.
The degree is 3.
Add the exponents of the variables 1 1.
The degree is 1.
Add the exponents of the variables 3 3.
The degree is 3.
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A polynomial is a monomial or a sum or difference
of monomials. The degree of a polynomial is the
degree of the term with the greatest degree.
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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.
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Additional Example 2A Writing Polynomials in
Standard Form
Write the polynomial in standard form. Then give
the leading coefficient.
6x 7x5 4x2 9
Find the degree of each term. Then arrange them
in descending order
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Additional Example 2B Writing Polynomials in
Standard Form
Write the polynomial in standard form. Then give
the leading coefficient.
y2 y6 - 3y
Find the degree of each term. Then arrange them
in descending order
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y5 1y5
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Check It Out! Example 2a
Write the polynomial in standard form. Then give
the leading coefficient.
16 4x2 x5 9x3
Find the degree of each term. Then arrange them
in descending order
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Check It Out! Example 2b
Write the polynomial in standard form. Then give
the leading coefficient.
18y5 3y8 14y
Find the degree of each term. Then arrange them
in descending order
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Some polynomials have special names based on
their degree and the number of terms they have.
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Additional Example 3 Classifying Polynomials
Classify each polynomial according to its degree
and number of terms.
A. 5n3 4n
5n3 4n is a cubic binomial.
Degree 3 Terms 2
B. 2x
2x is a linear monomial.
Degree 1 Terms 1
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Check It Out! Example 3
Classify each polynomial according to its degree
and number of terms.
a. x3 x2 x 2
x3 x2 x 2 is a cubic polynomial.
Degree 3 Terms 4
b. 6
Degree 0 Terms 1
3y8 18y5 14y is an 8th-degree trinomial.
c. 3y8 18y5 14y
Degree 8 Terms 3
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Additional Example 4 Application
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?
Substitute the time for t to find the lip balms
height.
16t2 220
16(3)2 200
The time is 3 seconds.
16(9) 200
Evaluate the polynomial by using the order of
operations.
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Additional Example 5 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
above the water.
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Check It Out! Example 4
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?
Substitute the time for t to find the fireworks
height.
16t2 400t 6
16(5)2 400(5) 6
The time is 5 seconds.
16(25) 400(5) 6
400 2000 6
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Check It Out! Example 4 Continued
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?
When the firework explodes, it will be 1606 feet
above the ground.
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A root of a polynomial in one variable is a value
of the variable for which the polynomial is equal
to 0.
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Additional Example 5 Identifying Roots of
Polynomials
Tell whether each number is a root of 3x2 48.
A. 4
B. 0
Substitute for x.
Simplify.
4 is a root of 3x2 48.
0 is not a root of 3x2 48.
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Additional Example 5 Identifying Roots of
Polynomials
Tell whether each number is a root of 3x2 48.
C. 4
3x2 48
3(4)2 48
Substitute for x.
3(16) 48
48 48
Simplify.
4 is a root of 3x2 48.
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Check It Out! Example 5
Tell whether 1 is a root of 3x3 x 4.
3x3 x 4
3(1)3 (1) 4
Substitute for x.
3(1) 1 4
3 1 4
Simplify.
1 is a root of 3x3 x 4.
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