44.6 Multivariable Polynomial Operations

Multivariable Polynomial Operations extends the addition and subtraction of polynomials to expressions involving more than one variable, requiring every variable and its exponent to match completely before terms can be combined, and adopting a consistent ordering convention for the resulting expression.

Complete Variable-Part Matching

Requirement

For two terms in a multivariable polynomial to be considered like terms, every single variable present in one term must also appear in the other, each raised to exactly the same exponent.

Example

Variable Exponent Pattern Preservation

Requirement

Even a single differing exponent among the shared variables prevents two terms from being considered like terms, regardless of how similar the terms otherwise appear.

Example

Multivariable Like-Term Combination

Procedure

Once two or more terms are confirmed to share an identical variable part, including every exponent, their coefficients are added or subtracted together, exactly as with single-variable like terms.

Example

Multivariable Unlike-Term Retention

Procedure

Any term whose variable pattern does not exactly match another term across the operands is carried forward into the final result unchanged, exactly as in single-variable polynomial operations.

Example

Continuing from the earlier example, $3x^{2}y$ and $5x{y}^2$ both remain as separate terms in any sum or difference involving them, since neither has a matching partner.

Multivariable Polynomial Sum

Example

Adding $4x^{2}y+3x{y}^2$ and $2x^{2}y-x{y}^2$ combines the matching $x^{2}y$ terms and the matching $x{y}^2$ terms separately:

Multivariable Polynomial Difference

Example

Subtracting $2x^{2}y-x{y}^2$ from $4x^{2}y+3x{y}^2$ first reverses the signs of the second polynomial, then combines matching terms:

Adopted Multivariable Term Ordering

Convention

When arranging a multivariable polynomial result, terms are conventionally ordered by their combined degree from highest to lowest, with a consistent tiebreaking rule, such as alphabetical priority of the first variable, applied whenever two terms share the same combined degree.

Example

In the result $6x^{2}y+2x{y}^2$, both terms share a combined degree of three, so the term with the higher exponent on $x$ is placed first, following the adopted ordering convention.