Monomial Lesson: Definition, Factoring and Examples

Lesson Overview

• What Is Monomial?
• How to Find a Monomial
• Factoring Monomials
• Difference Between Monomials, Binomials, And Trinomials
• Practice Problems and Examples

A monomial is a basic concept in algebra and serves as the foundation for more complex equations. Monomials can include numbers, variables, or a combination of both, connected by multiplication. 

For example, 3x, 7y squared, and 12 are all monomials. They are simple but useful tools in math, especially in equations, functions, and polynomials.

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Sergeant Math 8 Multiplying Monomials

What Is Monomial?

A monomial is a single algebraic term that can be a constant, a variable, or a product of constants and variables with whole number exponents. 

Monomials do not have addition, subtraction, or division involving variables.

Examples:

• Constants: 7, -3, ½
• Variables: x, y, z

Products: 2x², -5xy, 4a²b³c

How to Find a Monomial

Monomials can be identified by analyzing the structure of algebraic expressions. Key characteristics include a single term with constants, variables, and whole number exponents. 

Degree of Monomial

The degree of a monomial is the sum of the exponents of its variables. This includes any implicit exponents of 1 for variables without explicitly written exponents.

Examples:

• 2xy³: The degree is 1 + 3 = 4 (remember that 'x' has an implicit exponent of 1).
• 5x²y⁴z: The degree is 2 + 4 + 1 = 7.

-8: The degree of a non-zero constant is 0.

Operations on Monomials

Monomials can be combined using standard arithmetic operations: addition, subtraction, multiplication, and division. These operations follow specific rules based on the properties of exponents and algebraic manipulation.

• Addition and Subtraction of Monomials

Monomials can only be added or subtracted if they have the same variable(s) raised to the same power(s).

Example: 2x² + 5x²

Solution:

• The terms have the same variable x², so they are like terms.
• Combine the coefficients (2 and 5): 2x² + 5x² = (2 + 5)x² = 7x²

Answer:
7x²

• Multiplication of Monomials

When multiplying monomials, multiply the coefficients and add the exponents of like variables.

Example: (3x³y)(2xy²)

Solution:

• Multiply the coefficients: 3 × 2 = 6.
• Multiply the variables with the same base:
  • For x, add the exponents: x³ × x¹ = x^(3+1) = x⁴.
  • For y, add the exponents: y¹ × y² = y^(1+2) = y³.

Answer:
6x⁴y³

• Division of Monomials 

When dividing monomials, divide the coefficients and subtract the exponents of like variables.

Example: (10x⁵y³) / (2x²y)

Solution:

• Divide the coefficients: 10 ÷ 2 = 5.
• Divide the variables:
  • For x, subtract the exponents: x⁵ / x² = x^(5-2) = x³.
  • For y, subtract the exponents: y³ / y¹ = y^(3-1) = y².

Answer:
5x³y²

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Monomials and Polynomials! Algebra Trivia Questions Quiz

Factoring Monomials

Factoring a monomial involves expressing it as a product of its prime factors, similar to factoring whole numbers. This process is done by factoring the coefficient and variables separately.

Example

Factorize the monomial 24y³.

1. Factor the coefficient: The prime factors of 24 are 8 and 3.
2. Factor the variable: y³ can be factored as y × y × y.

Therefore, the complete factorization of 24y³ is 2 × 2 × 2 × 3 × y × y × y

Factoring monomials is a useful technique in simplifying expressions and identifying common factors within algebraic terms.

Difference Between Monomials, Binomials, And Trinomials

Algebraic expressions are classified by their number of terms. Monomials have one term, binomials have two, and trinomials have three.

Practice Problems and Examples

1. Identify the Monomials: Determine which of the following expressions are monomials:

• 5x³
• 2x + y
• -7a²b³c
• 4/x
• √x

Solution:

• 5x³ is a monomial (single term, non-negative integer exponents, no variable in the denominator).
• 2x + y is not a monomial (it has two terms – it's a binomial).
• -7a²b³c is a monomial.
• 4/x is not a monomial (variable in the denominator).
• √x is not a monomial (the exponent of x is ½, which is not an integer).

2. Determine the Degree: Find the degree of each monomial:

• 8xy²
• -3p³q⁵r
• 12

Solution:

• The degree of 8xy² is 1 + 2 = 3.
• The degree of -3p³q⁵r is 3 + 5 + 1 = 9.
• The degree of 12 is 0 (non-zero constant).

3. Perform the Operations: Simplify the following expressions:

• 3x²y + 5x²y
• (4a³b²)(2ab⁴)
• (12m⁵n⁴) / (3mn²)

Solution:

• 3x²y + 5x²y = 8x²y
• (4a³b²)(2ab⁴) = 8a⁴b⁶
• (12m⁵n⁴) / (3mn²) = 4m⁴n²

4. Factor the Monomial: Completely factor the monomial 24a³b².

Solution:

• Factor the coefficient: 24 = 2 × 2 × 2 × 3
• Factor the variables: a³b² = a × a × a × b × b

Therefore, the complete factorization of 24a³b² is 2 × 2 × 2 × 3 × a × a × a × b × b.

5. Application Problem: The area of a rectangle is given by the monomial 18x²y³. If the length of the rectangle is 6xy, what is the width?

Solution:

• Recall that the area of a rectangle is length × width.
• To find the width, divide the area by the length: (18x²y³) / (6xy) = 3xy²

Therefore, the width of the rectangle is 3xy².

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Multiplying and Dividing Monomials