Notes on Multiplication and Division of Rational Algebraic Expressions

Objectives

• Recall multiplication and division of fractions.
• Multiply and divide rational algebraic expressions.

Multiplying Fractions

1. Process: Multiply both the numerators and denominators.
   • Example:
     • If ( \frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d} )
     • Important: ( b \neq 0 ) and ( d \neq 0 ) to avoid undefined fractions.
2. Examples:
   • ( \frac{2}{3} \times \frac{3}{4} = \frac{2 \times 3}{3 \times 4} = \frac{6}{12} = \frac{1}{2} )
   • ( \frac{4}{5} \times \frac{2}{4} = \frac{8}{20} = \frac{2}{5} ) (after simplification)
   • Cancellations: Cancel common factors before multiplying.

Multiplying Rational Algebraic Expressions

1. Process: Similar to fractions.
   • Example:
     • If ( \frac{p}{q} \times \frac{r}{s} = \frac{p \times r}{q \times s} )
     • Important: ( q \neq 0 ) and ( s \neq 0 ), and ( p, q, r, s ) are polynomials.
2. Steps:
   • Factor the numerator and denominator.
   • Cancel out common factors.
   • Multiply the remaining terms and simplify.
3. Example:
   • ( \frac{a^5}{10} \times \frac{5}{a^3} )
   • Result: ( \frac{5a^5}{10a^3} = \frac{5}{10}a^{5-3} = \frac{1}{2}a^2 )
4. Further Example:
   • ( \frac{30b^2}{60} \times \frac{4c^2}{15b^4} )
   • Multiply: ( \frac{120b^2c^2}{90b^4} )
   • Simplified Result: ( \frac{4c^2}{3b^2} )

Dividing Rational Algebraic Expressions

1. Process: Get the reciprocal of the second fraction (divisor).
   • Example:
     • ( \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} )
2. Steps:
   • Factor the numerator and denominator.
   • Cancel out common factors.
   • Multiply the remaining terms.
3. Example:
   • ( \frac{a^2 - 9}{a - 3} \div \frac{a + 3}{3a} )
   • Resulting in: ( \frac{(a + 3)(a - 3)}{(a - 3)(3a)} )
   • Cancel common factors: ( (a - 3) ) to yield ( \frac{(a + 3)}{3} )
4. Further Example:
   • ( \frac{x^2 + 2x - 8}{3a} \div \frac{x^2 - 16}{3} )
   • Factor: ( (x + 4)(x - 2) \div (x + 4)(x - 4) )
   • Cancel common factors to yield final answer.

Summary

• Various methods for multiplying and dividing rational algebraic expressions.
• Importance of cancelling common factors and simplifying results.
• Always check for undefined expressions by ensuring denominators are not zero.

Conclusion

• Understanding these processes will aid in solving algebraic expressions effectively.