Multi-Variable Expressions

A multi-variable expression is an algebraic expression that contains two or more distinct variables, each representing an independent unknown quantity. For example, 3x²y − 5xy² + 7z is a polynomial in three variables: x, y, and z. Unlike single-variable polynomials, multi-variable expressions require you to track each variable separately, assign each one a value when evaluating, and apply the rules of exponents and arithmetic across every term. These expressions appear throughout mathematics, physics, economics, and computer science whenever a quantity depends on more than one changing factor. The degree of a term in a multi-variable polynomial is the sum of all the exponents on the variables in that term. For instance, the term 4x³y²z has degree 3 + 2 + 1 = 6. The degree of the entire polynomial is the highest degree among all its terms. Identifying degrees is important because it tells you about the behavior of the expression and guides decisions about how to simplify or factor it. Like terms must share exactly the same variables raised to exactly the same powers — so 2x²y and 5x²y are like terms (combinable), but 2x²y and 2xy² are not. Evaluating a multi-variable expression means substituting specific numerical values for each variable and computing the result using the correct order of operations: parentheses, exponents, multiplication and division, then addition and subtraction (PEMDAS). Manipulation — including addition, subtraction, multiplication, and factoring — follows the same algebraic laws (commutative, associative, distributive) that govern single-variable expressions, but demands careful bookkeeping since many different variable combinations can appear simultaneously. Mastering multi-variable expressions is an essential step toward understanding functions of several variables, systems of equations, and multivariable calculus.

Key takeaways

Identify Every Term and Its Degree
Collect Like Terms
Apply the Distributive Property for Multiplication
Evaluate by Substitution
Factor Multi-Variable Polynomials

Identify Every Term and Its Degree

Write out each term separated by + or − signs. For each term, find the degree by adding the exponents of all variables. Example: in 5x²y³ − 3xy + 8y², the degrees are 5, 2, and 2 respectively, so the polynomial has degree 5. Labeling degrees first prevents errors in later steps.

Collect Like Terms

Two terms are 'like' only if they have identical variable parts — the same variables raised to the same powers. Coefficients may differ, but variable parts must match exactly. Example: 7x²y + 3x²y = 10x²y, but 7x²y + 3xy² cannot be combined because the variable parts x²y and xy² are different.

Apply the Distributive Property for Multiplication

To multiply two multi-variable polynomials, distribute every term in the first factor across every term in the second factor, multiply coefficients together, and add exponents of matching bases. For example: (2x + y)(x − 3y) = 2x·x + 2x·(−3y) + y·x + y·(−3y) = 2x² − 6xy + xy − 3y² = 2x² − 5xy − 3y².

Evaluate by Substitution

Replace each variable with its given numerical value. Use parentheses around each substituted number to avoid sign errors, then simplify using order of operations. If x = 2 and y = −1, then 2x² − 5xy − 3y² = 2(2)² − 5(2)(−1) − 3(−1)² = 2(4) + 10 − 3(1) = 8 + 10 − 3 = 15.

Factor Multi-Variable Polynomials

Look first for a greatest common factor (GCF) across all terms. For example, 6x²y − 9xy² = 3xy(2x − 3y). Then check whether the remaining factor matches a special pattern such as difference of squares: a² − b² = (a+b)(a−b), which works with multiple variables too — for instance, 4x² − 9y² = (2x + 3y)(2x − 3y).

The Key Equation

P(x, y) = a_{mn}x^m y^n + \cdots + a_{10}x + a_{01}y + a_{00}

P(x, y) — Polynomial function (dimensionless): The value of the polynomial for specific inputs x and y.
a_{mn} — Coefficient (dimensionless): A real-number constant multiplying the term x^m y^n.
m, n — Exponents (non-negative integers): The powers to which x and y are raised in a given term; their sum m+n is the degree of that term.
x, y — Variables (dimensionless): Independent unknowns that can each take any real-number value.

Worked Example

Problem. Given the expression E = 2x²y − 3xy² + x − 4y + 5, evaluate E when x = 3 and y = −2. Then determine the degree of E.

E = -56

Answer. E = −56 when x = 3 and y = −2; the polynomial has degree 3.

Figure

Diagram

Origins · 1 minFrançois Viète and how this was discovered

François Viète

1540–1603 · French

Viète introduced the systematic use of letters to represent both known and unknown quantities in algebra, laying the foundation for symbolic algebra as we know it. His 1591 work 'In Artem Analyticam Isagoge' was the first text to use variables (letters) consistently, which made it possible to write and manipulate multi-variable expressions in general form rather than working only with specific numbers. Viète worked as a lawyer and royal privy councillor for King Henry III of France; mathematics was essentially his hobby. During wartime, he spent several years decrypting complex Spanish ciphers for the French crown — applying the same systematic analytical thinking he used in algebra.

Single-Variable Polynomials — Single-variable polynomials are the foundation for multi-variable ones; all the rules for degree, like terms, and operations extend directly to the multi-variable case.
Factoring Polynomials — Factoring techniques like GCF extraction and difference of squares apply directly to multi-variable expressions and are a key manipulation skill.
Substitution and Evaluation — Evaluating multi-variable expressions is an application of substitution, and the same order-of-operations rules must be followed carefully with multiple variables.
Systems of Equations — Systems of equations involve multiple variables simultaneously and rely on the ability to manipulate and evaluate multi-variable expressions.