13.5 Variable and Monomial Factor Expansion

Variable and Monomial Factor Expansion applies the distributive property when the outer multiplying factor is itself a variable or a monomial containing a coefficient and one or more variables, producing new terms in which variable factors from the outer monomial combine with the variable factors of each grouped term.

Distributing a Bare Variable Factor

Variable Factor Applied to a Sum

When a single variable multiplies a grouped sum, that variable is multiplied by each term inside the group individually, producing new terms in which the variable becomes an additional factor alongside whatever the original term already contained.

Variable Factor Applied to a Difference

When a single variable multiplies a grouped difference, that variable is multiplied by each of the two terms individually, producing new terms joined by the same subtraction as in the original group.

Distributing a Monomial with a Coefficient

Numerical Coefficient with a Variable Factor

When the outer factor is a monomial consisting of a numerical coefficient multiplied by a variable, that entire monomial is distributed across the group, with the coefficient scaling each generated product and the variable becoming an additional factor in each one.

Signed Monomial Outer Factor

When the monomial outer factor carries a negative coefficient, every generated product has its sign flipped, in addition to receiving the monomial's variable factor and the magnitude of its coefficient, exactly as with any negative outer factor.

Combining Variable Factors within Each Product

Variable Product Formation

When the outer variable factor multiplies a term that already contains a different variable, the two variables combine as separate multiplied factors within the new term, forming a multivariable product.

Repeated Variable Factor Formation

When the outer variable factor multiplies a term that already contains the same variable, the two occurrences of that variable combine using exponent rules, with their exponents added together to form a single powered factor in the new term.

Multivariable Monomial Expansion

When the outer factor is a monomial containing more than one variable, every one of those variables is carried into each generated product, combining with whatever variables the corresponding grouped term already contained, following the same combination rules for matching and differing variables.

Organizing the Distribution Process

Coefficient and Variable-Part Distribution

Distributing a monomial factor is most reliably carried out by handling the numerical coefficient and each variable factor of the monomial separately for every term in the group: multiplying the coefficients first, then combining each shared variable using exponent addition, and carrying forward any variable that appears in only one of the two factors unchanged.

Confirming the Expansion

Monomial Expansion Verification

A completed monomial expansion can be verified by substituting chosen numerical values for every variable involved into both the original factored expression and the expanded expression, confirming that both produce the same result, which catches errors in coefficient multiplication, sign handling, or variable combination during the distribution.