Math Rx Module 6.3 Rational Expressions Worksheet A rational expression is a fraction with polynomials: ܲ ‫ܲ| { = ݊݋݅ݏݏ݁ݎ݌ݔ݁ ݈ܽ݊݋݅ݐܽݎ‬, ܳ ܽ‫ ≠ ܳ ;ݏ݈ܽ݅݉݋݊ݕ݈݋݌ ݁ݎ‬0} ܳ Because Q (the denominator) cannot be 0, we need to be aware of restrictions on the variables. That is any value for the variable that would make the denominator = 0. To find this, set the denominator = 0 and solve. The solutions will be the restrictions on the variable. Strategy Example To simplify a rational expression: ‫ ݔ‬ଶ − 7‫ ݔ‬+ 12 = 1. Factor the numerator and denominator ሺ‫ ݔ‬ଶ − 9ሻ 2. Cancel like factors ሺ‫ ݔ‬− 3ሻሺ‫ ݔ‬− 4ሻ ‫ݔ‬−4 = ሺ‫ ݔ‬− 3ሻሺ‫ ݔ‬+ 3ሻ ‫ݔ‬+3 To multiply rational expressions: ‫ ݔ‬ଶ − 16 2‫ ݔ‬ଶ − 7‫ ݔ‬+ 6 • = 1. Factor the numerators and denominators of ሺ2‫ ݔ‬ଶ − ‫ ݔ‬− 3ሻ 2‫ ݔ‬+ 8 each “fraction” 2. Cancel like factors among all “fractions” ሺ‫ ݔ‬+ 4ሻሺ‫ ݔ‬− 4ሻ ሺ2‫ ݔ‬− 3ሻሺ‫ ݔ‬− 2ሻ • = 3. You do not have to re-multiply your factors ሺ2‫ ݔ‬− 3ሻሺ‫ ݔ‬+ 1ሻ 2ሺ‫ ݔ‬+ 4ሻ together at the end. ሺ‫ ݔ‬− 4ሻሺ‫ ݔ‬− 2ሻ 2ሺ‫ ݔ‬+ 1ሻ To divide rational expressions: 5‫ ݔ‬+ 5 ‫ ݔ‬ଶ + 2‫ ݔ‬+ 1 ÷ = 1. Change division to multiplication (multiply ‫ ݔ‬ଶ − 12‫ ݔ‬+ 36 3‫ ݔ‬− 18 by the reciprocal of the second fraction) 2. Multiply by factoring and cancelling. 5‫ ݔ‬+ 5 3‫ ݔ‬− 18 • ଶ = ଶ ‫ ݔ‬− 12‫ ݔ‬+ 36 ‫ ݔ‬+ 2‫ ݔ‬+ 1 5ሺ‫ ݔ‬+ 1ሻ 3ሺ‫ ݔ‬− 6ሻ • = ሺ‫ ݔ‬− 6ሻሺ‫ ݔ‬− 6ሻ ሺ‫ ݔ‬+ 1ሻሺ‫ ݔ‬+ 1ሻ 15 ሺ‫ ݔ‬− 6ሻሺ‫ ݔ‬+ 1ሻ ܴܾ݁݉݁݉݁‫ݐ ݎ‬ℎܽ‫ = ݋ݐ ݁݀݅ݒ݅݀ ݏ݁ݐ݅ݏ݋݌݌݋ ݐ‬−1 ‫ݔ‬−1 = −1 1−‫ݔ‬ The restriction on the variable in the simplified version is ‫ ≠ ݔ‬−9 because if ‫ ݏܽݓ ݔ‬− 9, that would cause the denominator to = 0. 6_3 Math Rx Rational Expression Ws.Docx ሺ9 − ‫ݔ‬ሻ = ሺ‫ ݔ‬ଶ − 81ሻ ሺ9 − ‫ݔ‬ሻ = ሺ‫ ݔ‬+ 9ሻሺ‫ ݔ‬− 9ሻ ሺ9 − ‫ݔ‬ሻ and ሺ‫ ݔ‬− 9ሻ are opposites. Rewrite in descending powers to see why. −1 1 =− ‫ݔ‬+9 ‫ݔ‬+9 Page 1 Math Rx Module 6.3 Rational Expressions Worksheet Reduce the following rational expressions to lowest terms by factoring and cancelling. Problem Work Answer 1. 12ܽ + 60 6ܽ + 30 2. ‫ ݔ‬ଶ − 7‫ ݔ‬+ 10 ‫ݔ‬ଶ − ‫ ݔ‬− 2 3. 2ܽଷ − 7ܽଶ + 5ܽ 2ܽଷ − 5ܽଶ + 3ܽ 4. 4‫ ݔ‬ଶ − 12‫ ݔ‬+ 9 4‫ ݔ‬− 6 5. ‫ݔ‬−2 ‫ݔ‬ଶ − 4 6_3 Math Rx Rational Expression Ws.Docx Page 2 Math Rx Module 6.3 Rational Expressions Worksheet Perform the indicated operation. Reduce your answer to lowest terms. Problem 6. ‫ ݕݔ‬+ 5‫ ݕ‬+ 5‫ ݔ‬+ 25 ‫ ݕݔ‬+ 3‫ ݔ‬+ 5‫ ݕ‬+ 15 7. ܽଶ + 10 + 25 ܽଶ − 25 ÷ 4ܽ − 4 ܽ−1 8. 2‫ ݔ‬ହ ‫ݔ‬−3 • 2‫ ݔ‬ଶ − 6‫ݔ‬ ‫ݔ‬ଷ 9. ‫ ݔ‬ଷ − 2‫ ݔ‬ଶ + ‫ ݔ‬− 2 5‫ ݔ‬ଶ + 4‫ ݔ‬− 1 • ଶ ‫ݔ‬ସ − 1 5‫ ݔ‬− 11‫ ݔ‬+ 2 10. Answer ‫ ݔ‬+ 1 4‫ ݔ‬+ 4 ÷ ‫ݔ‬ଶ − 9 ‫ ݔ‬+ 3 6_3 Math Rx Rational Expression Ws.Docx Page 3