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Monomials and Polynomials: Definition, Examples, and Degree, Slides of Algebra

Definitions, examples, and explanations of monomials and polynomials, including their degrees. Monomials are terms with one variable or constant, while polynomials are sums or differences of monomials. Examples of both and explains how to determine the degree of a polynomial.

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2021/2022

Uploaded on 09/12/2022

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Monomial Definition: (the prefix “mono” means one) a monomial is a constant or variable or a
multiple of such (i.e., just one term) Variables can only have exponents that are non-negative
integers.
Examples of monomials:
2𝑥 −3𝑥𝑦 1
2𝑥2 −5 7𝑥4
Polynomial Definition: (the prefix “poly” means many)
A sum or difference of monomials. The exponent of each variable must be a non-negative
integer.
Examples of Polynomials:
2𝑥 4 6𝑥2 7𝑥 + 2 𝑥3 2𝑥 2+11𝑥 1
Examples of NON-Polynomials:
log(𝑥 + 5) 2𝑥+ 5 𝑥2+1
𝑥 𝑥 + 5𝑥 3 6𝑥3+𝑥1
2+ 5 |𝑥 + 5|
The DEGREE of a polynomial is the highest exponent.
For example, the following polynomials have the following DEGREES:
7 has a degree of 0 (because 7 𝑥0= 7 1= 7)
2𝑥 4 has a degree of 1
6𝑥2 7𝑥 + 2 has a degree of 2
𝑥3 2𝑥2 has a degree of 3
𝑥 + 5𝑥3 2𝑥 410𝑥2 has a degree of 4
Standard Form of a Polynomial:
Polynomials should always be written in standard form. Standard form is when each term is written in
descending order of its exponent.
For example, 3𝑥 7𝑥3+ 9 2𝑥4 𝑥2 is NOT in standard form.
It should be written as: −2𝑥4 7𝑥3 𝑥 2+ 3𝑥 + 9
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Monomial Definition: (the prefix “mono” means one) a monomial is a constant or variable or a

multiple of such (i.e., just one term) Variables can only have exponents that are non-negative

integers.

Examples of monomials: 2 𝑥 − 3 𝑥𝑦 1 2

𝑥^2 − 5 7 𝑥^4

Polynomial Definition: (the prefix “poly” means many)

A sum or difference of monomials. The exponent of each variable must be a non-negative

integer.

Examples of Polynomials: 2 𝑥 − 4 6 𝑥^2 − 7 𝑥 + 2 𝑥^3 − 2 𝑥^2 + 11 𝑥 − 1 Examples of NON-Polynomials:

log(𝑥 + 5 )^2 𝑥^ + 5 𝑥^2 +

1 𝑥

√𝑥 + 5 𝑥 − 3 6 𝑥^3 +𝑥

1

The DEGREE of a polynomial is the highest exponent. For example, the following polynomials have the following DEGREES: 7 has a degree of 0 (because 7 ∙ 𝑥^0 = 7 ∙ 1 = 7 ) 2 𝑥 − 4 has a degree of 1 6 𝑥^2 − 7 𝑥 + 2 has a degree of 2 𝑥^3 − 2 𝑥^2 has a degree of 3 𝑥 + 5 𝑥^3 − 2 𝑥^4 − 10 𝑥^2 has a degree of 4 Standard Form of a Polynomial: Polynomials should always be written in standard form. Standard form is when each term is written in descending order of its exponent. For example, 3 𝑥 − 7 𝑥^3 + 9 − 2 𝑥^4 − 𝑥^2 is NOT in standard form. It should be written as: − 2 𝑥^4 − 7 𝑥^3 − 𝑥^2 + 3 𝑥 + 9

Polynomials can be named by the number of terms and/or by the degree.

Polynomial Degree Name by Degree

18 0 constant

½x - 5 1 linear

x² + 2x – 7 2 quadratic

4 x³ - 10 x + 6 3 cubic

− 2 x^4 − 7x^3 − x^2 4 No special name^ 

We just say “a polynomial with degree 4”

Polynomial Number of

Terms

Name by Terms

18 1 monomial

½x - 5 2 binomial

x² + 2x – 7 3 trinomial

4 x³ + x²- 10 x + 6 4 No special name^ ^ So, we just

call it a polynomial.

Determine if the following expressions are polynomials: 1 4 𝑥^2 + 10 𝑥 yes 4 𝑥 +^10 𝑥^ no^ (x cannot be in the denominator) − 2 𝑥^3 + 𝑥^2 − 7 𝑥 + 1 yes 6 𝑥^3 − 9 𝑥 + 𝑥 1 (^2) + 5 no (the exponent of x can only be 0, 1, 2, 3, 4, …etc) 2 𝑥^3 + 𝑥−^2 − 4 𝑥 + 3 no (the exponent of x can only be 0, 1, 2, 3, 4, …etc) 7 𝑥^ + 1 no (it’s an exponential) 𝑥^7 + 1 yes 𝑥 + 8 1 (^3) yes (it’s the same as x + 2, so it’s linear) |𝑥| + 5 no (it’s an absolute value function) | 5 | + 𝑥 yes (it’s the same as 5 + x, so it’s linear) 4 + 𝑙𝑜𝑔 2 𝑥 no (it’s logarithmic) 𝑥^2 + 3 𝑥 + 𝑙𝑜𝑔 24 yes (it’s the same as 𝑥^2 + 3 𝑥 + 2 , so it’s a quadratic)