In algebra, polynomials are expressions consisting of variables and coefficients, combined through addition, subtraction, multiplication, and exponentiation operations. When dealing with multivariate polynomials, which involve multiple variables, factoring out a monomial is a common technique used to simplify and manipulate expressions. We’ll explore the process of factoring out a monomial from multivariate polynomials, providing step-by-step guidance and examples to elucidate the concept.

### Understanding Multivariate Polynomials

Multivariate polynomials are polynomial expressions that involve more than one variable. For example, \(3x^2y + 2xy^2 – 5xyz\) is a multivariate polynomial with three terms and three variables: \(x\), \(y\), and \(z\). These polynomials can be manipulated using various algebraic techniques, including factoring, expanding, and simplifying.

### What is a Monomial?

A monomial is a polynomial expression consisting of a single term. It is characterized by a coefficient and one or more variables raised to non-negative integer exponents. For example, \(2x^3\), \(5y\), and \(7z^2\) are all examples of monomials.

Factoring Out a Monomial from Multivariate Polynomials

Factoring out a monomial from a multivariate polynomial involves extracting a common factor that is present in each term of the polynomial. The process is similar to factoring out a monomial from a univariate polynomial but requires careful consideration of each variable’s exponent in the expression.

**Step-by-Step Process:**

- Identify the Monomial: Examine each term of the multivariate polynomial to identify a common monomial factor present in all terms. This monomial factor will have the smallest exponent for each variable across all terms.
- Determine the Common Factor: Once you’ve identified the monomial, determine its coefficient and variable factors. The coefficient is the numerical factor common to all terms, while the variable factors consist of the variables raised to their respective smallest exponents.
- Divide Each Term: Divide each term of the polynomial by the common factor identified in step 2. This involves dividing the coefficient of each term by the common coefficient and dividing each variable factor by the corresponding variable factors in the common monomial.
- Write the Factored Form: Write the factored form of the polynomial by placing the common factor outside parentheses and the resulting simplified terms inside parentheses. This represents the polynomial in factored form, with the common monomial factored out.

**Example:**

Consider the multivariate polynomial \(6x^2y + 9xy^2 – 12x^2y^2\).

1. Identify the Monomial: The common monomial factor among all terms is \(3xy\), as it has the smallest exponent for each variable (\(x\), \(y\)).

2. Determine the Common Factor: The common factor consists of the coefficient 3 and the variables \(x\) and \(y\) raised to the first power (\(x^1\) and \(y^1\)).

3. Divide Each Term:

– \(6x^2y\) Ã· \(3xy = 2x\)

– \(9xy^2\) Ã· \(3xy = 3y\)

– \(-12x^2y^2\) Ã· \(3xy = -4xy\)

4. Write the Factored Form:

Factored form: \(3xy(2x + 3y – 4xy)\)

**Applications and Importance:**

Factoring out a monomial from multivariate polynomials is a crucial algebraic technique used in various mathematical applications, including simplifying expressions, solving equations, and identifying patterns in data analysis. By factoring out common factors, mathematicians can reduce complex expressions to simpler forms, facilitating further analysis and manipulation.

Additionally, factoring out a monomial plays a fundamental role in polynomial division, polynomial factorization, and polynomial manipulation techniques such as partial fraction decomposition and polynomial long division. Mastery of this technique is essential for students studying algebra, calculus, and higher mathematics, as it forms the basis for many advanced mathematical concepts and problem-solving strategies.

Factoring out a monomial from multivariate polynomials is a fundamental algebraic technique used to simplify and manipulate polynomial expressions. By identifying common factors present in each term of a polynomial and factoring them out, mathematicians can reduce complex expressions to simpler forms, aiding in further analysis and problem-solving. Understanding this technique is essential for students and practitioners of mathematics, as it underpins many advanced mathematical concepts and techniques across various disciplines.