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 $\frac{x^{2} - 4 x + 4}{x^{2} - 4}$ .

<< Chapter < Page Page > Chapter >>

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

100% Free Mobile Applications
Receive real-time job alerts and never miss the right job again

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

Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra ii for the community college' conversation and receive update notifications?

Ask

	1 Neuroanatomy 01 The Cranial Nerves By Stephen Voron Start Quiz
	Journalism Midterm By Sheila Lopez Start Exam
	22 Biology 22 Prokaryotes Bacteria and Archaea MCQ By OpenStax Start Quiz
	9 AP 09 Joints MCQ Quiz By OpenStax Start Quiz
	9 BOD- Liver Quiz .... By Brooke Delaney Start Quiz
©flickr: Iqbal	Liver Cancer By Darlene Paliswat Start Test
©flickr: Hey	Anatomy Physiology Final By Joli Julianna Start Test
	Cardiac Electrophysiology Basic By Mistry Bhavesh Start Quiz
	Social Work midterm By Katy Pratt Start Exam
	23 AP Key Terms 23 The Digestive System By OpenStax Start Key Terms