13.2 Positive Numerical Factor Expansion

Positive Numerical Factor Expansion applies the distributive property in the simplest case, where a positive number multiplies a grouped sum or difference, producing an expanded form in which every generated product keeps the original sign of the term it came from.

Expanding over Two Terms

Positive Factor over Two Added Terms

When a positive factor multiplies a group containing two terms joined by addition, the factor is multiplied by each of the two terms separately, and the two resulting products are added together in the expanded form.

Positive Factor over Two Subtracted Terms

When a positive factor multiplies a group containing two terms joined by subtraction, the factor is multiplied by each of the two terms separately, and the second resulting product is subtracted from the first in the expanded form, keeping the original subtraction in place.

Expanding over More Terms

Positive Factor over Several Terms

When a positive factor multiplies a group containing more than two terms, the factor is multiplied by every term in the group individually, with each resulting product keeping the sign that its original term carried inside the group.

Computing Each Generated Product

Numerical Product Calculation during Expansion

When the term being multiplied is itself a constant, the product is found by ordinary multiplication of the two numbers, producing a new constant value that becomes the constant term of the expanded expression.

Variable-Term Product Generation

When the term being multiplied contains a variable, the positive factor multiplies the coefficient of that term, while the variable part is carried through unchanged into the resulting product term.

Constant-Term Product Generation

When the term being multiplied is a constant with no variable, the resulting product is simply the numerical product of the two numbers, contributing only to the constant term of the fully expanded expression rather than to any variable term.

Completing the Expansion

Grouping Removal after Complete Distribution

Once the factor has been multiplied by every single term inside the group, the parentheses that originally enclosed the group are removed entirely, since the group no longer needs to be treated as a single unit; the expanded expression consists only of the separate generated products joined by their appropriate signs.

Confirming the Expansion

Positive-Factor Expansion Verification

A completed positive-factor expansion can be verified by substituting a chosen numerical value for any variables into both the original factored expression and the expanded expression, and confirming that both produce the same result; a mismatch signals that one of the terms inside the group was multiplied incorrectly or skipped during the distribution.