13.4 Fractional and Decimal Factor Expansion

Fractional and Decimal Factor Expansion applies the distributive property when the outer multiplying factor is a fraction or a decimal, requiring the same term-by-term multiplication as with whole-number factors but carried out using fraction or decimal arithmetic for each generated product.

Distributing a Fractional Factor

Unit Fraction as an Outer Factor

When the outer factor is a unit fraction, such as one over some whole number, distributing it across a group divides each term inside the group by that whole number, since multiplying by a unit fraction is equivalent to dividing by its denominator.

General Fractional Outer Factor

When the outer factor is a general fraction with a numerator other than one, distributing it across a group multiplies each term inside the group by that entire fraction, following the same term-by-term process used for any other numerical factor.

Computing Each Fractional Product

Fraction Multiplication during Expansion

Each product generated during a fractional expansion is computed using the standard rule for multiplying fractions: the fraction's numerator multiplies the term's numerator, or the whole-number term itself, while the fraction's denominator remains the divisor of the resulting product.

Fraction Reduction after Distribution

After multiplying the fractional factor by each term, any resulting fraction should be reduced to its simplest form, canceling common factors between the numerator and denominator of each generated product before the expanded expression is considered complete.

Distributing a Decimal Factor

Decimal Outer Factor

When the outer factor is a decimal, distributing it across a group multiplies each term inside the group by that decimal value, following standard decimal multiplication for each resulting product.

Signed Fractional and Decimal Factors

Signed Fractional Factor

When the fractional outer factor is negative, every generated product has its sign flipped in addition to being multiplied by the fraction's magnitude, exactly as with any other negative outer factor.

Signed Decimal Factor

When the decimal outer factor is negative, every generated product likewise has its sign flipped in addition to being scaled by the decimal's magnitude, following the same sign behavior as any other negative outer factor.

Choosing between Exact and Decimal Forms

Exact and Decimal Expansion Forms

An expansion involving a fractional factor can be left in exact fractional form or converted to a decimal expansion, and the choice depends on the context: exact fractional form avoids any rounding, while decimal form may be more convenient for further calculations expressed in decimals, provided any necessary rounding is applied consistently.

Confirming the Expansion

Fractional-Factor Result Verification

A completed fractional or decimal expansion can be verified by substituting a chosen numerical value for any variables into both the original factored expression and the expanded expression, confirming both produce the same result; this check is especially useful for catching errors introduced during fraction multiplication or reduction.