Section 1.6 : Rational Expressions
For problems 1 – 6 reduce each of the following to lowest terms.
- \( \displaystyle \frac{{{x^3} + 10{x^2}}}{{{x^2} + 6x - 40}}\)
- \( \displaystyle \frac{{{x^2} + 18x + 72}}{{2{x^2} + 11x - 6}}\)
- \( \displaystyle \frac{{{x^2} - 3x - 28}}{{49 - {x^2}}}\)
- \( \displaystyle \frac{{6{x^2} + 13x + 5}}{{3{x^2} + 26x + 35}}\)
- \( \displaystyle \frac{{ - {x^2} + 10x - 9}}{{ - {x^2} + 6x + 27}}\)
- \( \displaystyle \frac{{{x^3} + {x^2} - 20x}}{{{x^4} - 12{x^3} + 36{x^2}}}\)
For problems 7 – 13 perform the indicated operation and reduce the answer to lowest terms.
- \( \displaystyle \frac{{{x^2} + 14x + 40}}{{{x^2} + 2x - 8}}\,\centerdot \,\frac{{{x^2} + 5x - 14}}{{{x^2} + 7x - 30}}\)
- \( \displaystyle \frac{{4{x^3} - {x^2} - 3x}}{{{x^2} - 10x + 25}}\,\centerdot \,\frac{{10 + 3x - {x^2}}}{{{x^4} - {x^3}}}\)
- \( \displaystyle \frac{{{x^2} + 5x - 24}}{{{x^2} - 5x + 4}} \div \frac{{{x^2} + x - 12}}{{x - 1}}\)
- \( \displaystyle \frac{{6{x^2} + {x^3} - {x^4}}}{{{x^2} - 4}} \div \frac{{3{x^3} - 9{x^2}}}{{{x^2} + 6x - 16}}\)
- \( \displaystyle \frac{{3{x^2} + 23x + 14}}{{{x^2} + 4x + 3}} \div \frac{{6{x^2} + 13x + 6}}{{{x^2} + 2x + 1}}\)
- \( \displaystyle \frac{{5{x^2} - 18x - 8}}{{\displaystyle \frac{{x - 4}}{{x + 6}}}}\)
- \( \displaystyle \frac{{\displaystyle \frac{2}{{x + 4}}}}{{\displaystyle \frac{{6{x^3} + 17{x^2}}}{{{x^2} + 3x - 4}}}}\)
For problems 14 – 22 perform the indicated operations.
- \( \displaystyle \frac{2}{{3{x^2}}} - \frac{1}{{4{x^7}}} + \frac{7}{{6{x^3}}}\)
- \( \displaystyle \frac{{2x}}{{x + 9}} - \frac{{x - 1}}{x}\)
- \( \displaystyle \frac{{x + 1}}{{x - 1}} + \frac{6}{{x - 7}}\)
- \( \displaystyle \frac{9}{{{x^2} - 4}} - \frac{{7x}}{{{x^2} - 4x + 4}}\)
- \( \displaystyle \frac{{2x + 1}}{{4{x^2} - 3x - 7}} - \frac{{x + 3}}{{x + 1}} + \frac{x}{{4x - 7}}\)
- \( \displaystyle \frac{3}{{6x - {x^2}}} - \frac{x}{{{x^2} - 5x - 6}}\)
- \( \displaystyle \frac{2}{{{x^2} - 4x - 12}} + \frac{{8x}}{{{x^2} + 12x + 20}}\)
- \( \displaystyle \frac{3}{{{x^2}}} + \frac{{x + 9}}{{{x^2} + 5x}} - \frac{2}{{{x^2} + 10x + 25}}\)
- \( \displaystyle \frac{1}{{x + 1}} - \frac{2}{{{{\left( {x + 1} \right)}^2}}} - \frac{3}{{{{\left( {x + 1} \right)}^3}}}\)