Rational expressions: multiplying and dividing rational expressions (Page 2/2)

Algebra ii for the community Page 2 / 2

$\begin{array}{l} (x - 3) \cdot \frac{4 x - 9}{x^{2} - 6 x + 9} . \\ \frac{(x - 3)}{1} \cdot \frac{4 x - 9}{(x - 3) (x - 3)} \\ \frac{4 x - 9}{x - 3} \end{array}$

$\begin{array}{l} \begin{array}{l} \frac{- x^{2} - 3 x - 2}{x^{2} + 8 x + 15} \cdot \frac{4 x + 20}{x^{2} + 2 x} . & Factor –1 from the first numerator . \\ \frac{- (x^{2} + 3 x + 2)}{x^{2} + 8 x + 15} \cdot \frac{4 x + 20}{x^{2} + 2 x} & Factor . \\ \frac{- (x + 1) (x + 2)}{(x + 3) (x + 5)} \cdot \frac{4 (x + 5)}{x (x + 2)} & Multiply . \end{array} \\ \begin{matrix} \frac{- 4 (x + 1)}{x (x + 3)} = \frac{- 4 x - 1}{x (x + 3)} & or & \frac{- 4 x - 1}{x^{2} + 3 x} \end{matrix} \end{array}$

Practice set a

Perform each multiplication.

$\frac{5}{3} \cdot \frac{6}{7}$

$\frac{10}{7}$

$\frac{a^{3}}{b^{2} c^{2}} \cdot \frac{c^{5}}{a^{5}}$

$\frac{c^{3}}{a^{2} b^{2}}$

$\frac{y - 1}{y^{2} + 1} \cdot \frac{y + 1}{y^{2} - 1}$

$\frac{1}{y^{2} + 1}$

$\frac{x^{2} - x - 12}{x^{2} + 7 x + 6} \cdot \frac{x^{2} - 4 x - 5}{x^{2} - 9 x + 20}$

$\frac{x + 3}{x + 6}$

$\frac{x^{2} + 6 x + 8}{x^{2} - 6 x + 8} \cdot \frac{x^{2} - 2 x - 8}{x^{2} + 2 x - 8}$

$\frac{{(x + 2)}^{2}}{{(x - 2)}^{2}}$

Division of rational expressions

To divide one rational expression by another, we first invert the divisor then multiply the two expressions. Symbolically, if we let $P, Q, R,$ and $S$ represent polynomials, we can write

$\frac{P}{Q} \div \frac{R}{S} = \frac{P}{Q} \cdot \frac{S}{R} = \frac{P \cdot S}{Q \cdot R}$

Sample set b

Perform the following divisions.

$\begin{array}{l} \frac{6 x^{2}}{5 a} \div \frac{2 x}{10 a^{3}} . & Invert the divisor and multiply . \\ \frac{\overset{3}{6} x^{2}}{5 a} \cdot \frac{\overset{2}{10} a^{\overset{2}{3}}}{2 x} = \frac{3 x \cdot 2 a^{2}}{1} = 6 a^{2} x \end{array}$

$\begin{array}{l} \frac{x^{2} + 3 x - 10}{2 x - 2} \div \frac{x^{2} + 9 x + 20}{x^{2} + 3 x - 4} & Invert and multiply . \\ \frac{x^{2} + 3 x - 10}{2 x - 2} \cdot \frac{x^{2} + 3 x - 4}{x^{2} + 9 x + 20} & Factor . \\ \frac{(x + 5) (x - 2)}{2 (x - 1)} \cdot \frac{(x + 4) (x - 1)}{(x + 5) (x + 4)} \\ \frac{x - 2}{2} \end{array}$

$\begin{array}{l} (4 x + 7) \div \frac{12 x + 21}{x - 2} . & Write 4 x + 7 as \frac{4 x + 7}{1} . \\ \frac{4 x + 7}{1} \div \frac{12 x + 21}{x - 2} & Invert and multiply . \\ \frac{4 x + 7}{1} \cdot \frac{x - 2}{12 x + 21} & Factor . \\ \frac{4 x + 7}{1} \cdot \frac{x - 2}{3 (4 x + 7)} = \frac{x - 2}{3} \end{array}$

Practice set b

Perform each division.

$\frac{8 m^{2} n}{3 a^{5} b^{2}} \div \frac{2 m}{15 a^{7} b^{2}}$

$20 a^{2} m n$

$\frac{x^{2} - 4}{x^{2} + x - 6} \div \frac{x^{2} + x - 2}{x^{2} + 4 x + 3}$

$\frac{x + 1}{x - 1}$

$\frac{6 a^{2} + 17 a + 12}{3 a + 2} \div (2 a + 3)$

$\frac{3 a + 4}{3 a + 2}$

Excercises

For the following problems, perform the multiplications and divisions.

$\frac{4 a^{3}}{5 b} \cdot \frac{3 b}{2 a}$

$\frac{6 a^{2}}{5}$

$\frac{9 x^{4}}{4 y^{3}} \cdot \frac{10 y}{x^{2}}$

$\frac{a}{b} \cdot \frac{b}{a}$

$\frac{2 x}{5 y} \cdot \frac{5 y}{2 x}$

$\frac{12 a^{3}}{7} \cdot \frac{28}{15 a}$

$\frac{16 a^{2}}{5}$

$\frac{39 m^{4}}{16} \cdot \frac{4}{13 m^{2}}$

$\frac{18 x^{6}}{7} \cdot \frac{1}{4 x^{2}}$

$\frac{9 x^{4}}{14}$

$\frac{34 a^{6}}{21} \cdot \frac{42}{17 a^{5}}$

$\frac{16 x^{6} y^{3}}{15 x^{2}} \cdot \frac{25 x}{4 y}$

$\frac{20 x^{5} y^{2}}{3}$

$\frac{27 a^{7} b^{4}}{39 b} \cdot \frac{13 a^{4} b^{2}}{16 a^{5}}$

$\frac{10 x^{2} y^{3}}{7 y^{5}} \cdot \frac{49 y}{15 x^{6}}$

$\frac{14}{3 x^{4} y}$

$\frac{22 m^{3} n^{4}}{11 m^{6} n} \cdot \frac{33 m n}{4 m n^{3}}$

$\frac{- 10 p^{2} q}{7 a^{3} b^{2}} \cdot \frac{21 a^{5} b^{3}}{2 p}$

$- 15 a^{2} b p q$

$\frac{- 25 m^{4} n^{3}}{14 r^{3} s^{3}} \cdot \frac{21 r s^{4}}{10 m n}$

$\frac{9}{a} \div \frac{3}{a^{2}}$

$3 a$

$\frac{10}{b^{2}} \div \frac{4}{b^{3}}$

$\frac{21 a^{4}}{5 b^{2}} \div \frac{14 a}{15 b^{3}}$

$\frac{9 a^{3} b}{2}$

$\frac{42 x^{5}}{16 y^{4}} \div \frac{21 x^{4}}{8 y^{3}}$

$\frac{39 x^{2} y^{2}}{55 p^{2}} \div \frac{13 x^{3} y}{15 p^{6}}$

$\frac{9 p^{4} y}{11 x}$

$\frac{14 m n^{3}}{25 n^{6}} \div \frac{32 m}{20 m^{2} n^{3}}$

$\frac{12 a^{2} b^{3}}{- 5 x y^{4}} \div \frac{6 a^{2}}{15 x^{2}}$

$\frac{- 6 b^{3} x}{y^{4}}$

$\frac{24 p^{3} q}{9 m n^{3}} \div \frac{10 p q}{- 21 n^{2}}$

$\frac{x + 8}{x + 1} \cdot \frac{x + 2}{x + 8}$

$\frac{x + 2}{x + 1}$

$\frac{x + 10}{x - 4} \cdot \frac{x - 4}{x - 1}$

$\frac{2 x + 5}{x + 8} \cdot \frac{x + 8}{x - 2}$

$\frac{2 x + 5}{x - 2}$

$\frac{y + 2}{2 y - 1} \cdot \frac{2 y - 1}{y - 2}$

$\frac{x - 5}{x - 1} \div \frac{x - 5}{4}$

$\frac{4}{x - 1}$

$\frac{x}{x - 4} \div \frac{2 x}{5 x + 1}$

$\frac{a + 2 b}{a - 1} \div \frac{4 a + 8 b}{3 a - 3}$

$\frac{3}{4}$

$\frac{6 m + 2}{m - 1} \div \frac{4 m - 4}{m - 1}$

$x^{3} \cdot \frac{4 a b}{x}$

$4 a b x^{2}$

$y^{4} \cdot \frac{3 x^{2}}{y^{2}}$

$2 a^{5} \div \frac{6 a^{2}}{4 b}$

$\frac{4 a^{3} b}{3}$

$16 x^{2} y^{3} \div \frac{10 x y}{3}$

$21 m^{4} n^{2} \div \frac{3 m n^{2}}{7 n}$

$49 m^{3} n$

$(x + 8) \cdot \frac{x + 2}{x + 8}$

$(x - 2) \cdot \frac{x - 1}{x - 2}$

$x - 1$

${(a - 6)}^{3} \cdot \frac{{(a + 2)}^{2}}{a - 6}$

${(b + 1)}^{4} \cdot \frac{{(b - 7)}^{3}}{b + 1}$

${(b + 1)}^{3} {(b - 7)}^{3}$

${(b^{2} + 2)}^{3} \cdot \frac{b - 3}{{(b^{2} + 2)}^{2}}$

${(x^{3} - 7)}^{4} \cdot \frac{x^{2} - 1}{{(x^{3} - 7)}^{2}}$

${(x^{3} - 7)}^{2} (x + 1) (x - 1)$

$(x - 5) \div \frac{x - 5}{x - 2}$

$(y - 2) \div \frac{y - 2}{y - 1}$

$(y - 1)$

${(y + 6)}^{3} \div \frac{{(y + 6)}^{2}}{y - 6}$

${(a - 2 b)}^{4} \div \frac{{(a - 2 b)}^{2}}{a + b}$

${(a - 2 b)}^{2} (a + b)$

$\frac{x^{2} + 3 x + 2}{x^{2} - 4 x + 3} \cdot \frac{x^{2} - 2 x - 3}{2 x + 2}$

$\frac{6 x - 42}{x^{2} - 2 x - 3} \cdot \frac{x^{2} - 1}{x - 7}$

$\frac{6 (x - 1)}{(x - 3)}$

$\frac{3 a + 3 b}{a^{2} - 4 a - 5} \div \frac{9 a + 9 b}{a^{2} - 3 a - 10}$

$\frac{a^{2} - 4 a - 12}{a^{2} - 9} \div \frac{a^{2} - 5 a - 6}{a^{2} + 6 a + 9}$

$\frac{(a + 2) (a + 3)}{(a - 3) (a + 1)}$

$\frac{b^{2} - 5 b + 6}{b^{2} - b - 2} \cdot \frac{b^{2} - 2 b - 3}{b^{2} - 9 b + 20}$

$\frac{m^{2} - 4 m + 3}{m^{2} + 5 m - 6} \cdot \frac{m^{2} + 4 m - 12}{m^{2} - 5 m + 6}$

$\frac{r^{2} + 7 r + 10}{r^{2} - 2 r - 8} \div \frac{r^{2} + 6 r + 5}{r^{2} - 3 r - 4}$

$\frac{2 a^{2} + 7 a + 3}{3 a^{2} - 5 a - 2} \cdot \frac{a^{2} - 5 a + 6}{a^{2} + 2 a - 3}$

$\frac{(2 a + 1) (a - 6) (a + 1)}{(3 a + 1) (a - 1) (a - 2)}$

$\frac{6 x^{2} + x - 2}{2 x^{2} + 7 x - 4} \cdot \frac{x^{2} + 2 x - 12}{3 x^{2} - 4 x - 4}$

$\frac{x^{3} y - x^{2} y^{2}}{x^{2} y - y^{2}} \cdot \frac{x^{2} - y}{x - x y}$

$\frac{x (x - y)}{1 - y}$

$\frac{4 a^{3} b - 4 a^{2} b^{2}}{15 a - 10} \cdot \frac{3 a - 2}{4 a b - 2 b^{2}}$

$\frac{x + 3}{x - 4} \cdot \frac{x - 4}{x + 1} \cdot \frac{x - 2}{x + 3}$

$\frac{x - 2}{x + 1}$

$\frac{x - 7}{x + 8} \cdot \frac{x + 1}{x - 7} \cdot \frac{x + 8}{x - 2}$

$\frac{2 a - b}{a + b} \cdot \frac{a + 3 b}{a - 5 b} \cdot \frac{a - 5 b}{2 a - b}$

$\frac{a + 3 b}{a + b}$

$\frac{3 a {(a + 1)}^{2}}{a - 5} \cdot \frac{6 {(a - 5)}^{2}}{5 a + 5} \cdot \frac{15 a + 30}{4 a - 20}$

$\frac{- 3 a^{2}}{4 b} \cdot \frac{- 8 b^{3}}{15 a}$

$\frac{2 a b^{2}}{5}$

$\frac{- 6 x^{3}}{5 y^{2}} \cdot \frac{20 y}{- 2 x}$

$\frac{- 8 x^{2} y^{3}}{- 5 x} \div \frac{4}{- 15 x y}$

$- 6 x^{2} y^{4}$

$\frac{- 4 a^{3}}{3 b} \div \frac{2 a}{6 b^{2}}$

$\frac{- 3 a - 3}{2 a + 2} \cdot \frac{a^{2} - 3 a + 2}{a^{2} - 5 a - 6}$

$\frac{- 3 (a - 2) (a - 1)}{2 (a - 6) (a + 1)}$

$\frac{x^{2} - x - 2}{x^{2} - 3 x - 4} \cdot \frac{- x^{2} + 2 x + 3}{- 4 x - 8}$

$\frac{- 5 x - 10}{x^{2} - 4 x + 3} \cdot \frac{x^{2} + 4 x + 1}{x^{2} + x - 2}$

$\frac{- 5 (x^{2} + 4 x + 1)}{(x - 3) {(x - 1)}^{2}}$

$\frac{- a^{2} - 2 a + 15}{- 6 a - 12} \div \frac{a^{2} - 2 a - 8}{- 2 a - 10}$

$\frac{- b^{2} - 5 b + 14}{3 b - 6} \div \frac{- b^{2} - 9 b - 14}{- b + 8}$

$\frac{- (b - 8)}{3 (b + 2)}$

$\frac{3 a + 6}{4 a - 24} \cdot \frac{6 - a}{3 a + 15}$

$\frac{4 x + 12}{x - 7} \cdot \frac{7 - x}{2 x + 2}$

$\frac{- 2 (x + 3)}{(x + 1)}$

$\frac{- 2 b - 2}{b^{2} + b - 6} \cdot \frac{- b + 2}{b + 5}$

$\frac{3 x^{2} - 6 x - 9}{2 x^{2} - 6 x - 4} \div \frac{3 x^{2} - 5 x - 2}{6 x^{2} - 7 x - 3}$

$\frac{3 (x - 3) (x + 1) (2 x - 3)}{2 (x^{2} - 3 x - 2) (x - 2)}$

$\frac{- 2 b^{2} - 2 b + 4}{8 b^{2} - 28 b - 16} \div \frac{b^{2} - 2 b + 1}{2 b^{2} - 5 b - 3}$

$\frac{x^{2} + 4 x + 3}{x^{2} + 5 x + 4} \div (x + 3)$

$\frac{(x + 4) (x - 1)}{(x + 3) (x^{2} - 4 x - 3)}$

$\frac{x^{2} - 3 x + 2}{x^{2} - 4 x + 3} \div (x - 3)$

$\frac{3 x^{2} - 21 x + 18}{x^{2} + 5 x + 6} \div (x + 2)$

$\frac{3 (x - 6) (x - 1)}{{(x + 2)}^{2} (x + 3)}$

Exercises for review

( [link] ) If $a < 0$ , then $| a | =$ .

( [link] ) Classify the polynomial $4 x y + 2 y$ as a monomial, binomial, or trinomial. State its degree and write the numerical coefficient of each term.

binomial; 2; 4, 2

( [link] ) Find the product: $y^{2} (2 y - 1) (2 y + 1)$ .

( [link] ) Translate the sentence “four less than twice some number is two more than the number” into an equation.

$2 x - 4 = x + 2$

( [link] ) Reduce the fraction $\frac{x^{2} - 4 x + 4}{x^{2} - 4}$ .

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Source: OpenStax, Algebra ii for the community college. OpenStax CNX. Jul 03, 2014 Download for free at http://cnx.org/content/col11671/1.1

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