8.4: Multiplying and Dividing Rational Expressions

Multiplication Of Rational Expressions

Rational expressions are multiplied together in much the same way that arithmetic fractions are multiplied together. To multiply rational numbers, we do the following:

Method for Multiplying Rational Numbers

1. Reduce each fraction to lowest terms.
2. Multiply the numerators together.
3. Multiply the denominators together.

Rational expressions are multiplied together using exactly the same three steps. Since rational expressions tend to be longer than arithmetic fractions, we can simplify the multiplication process by adding one more step.

Method for Multiplying Rational Expressions

1. Factor all numerators and denominators.
2. Reduce to lowest terms first by dividing out all common factors. (It is perfectly legitimate to cancel the numerator of one fraction with the denominator of another.)
3. Multiply numerators together.
4. Multiply denominators. It is often convenient, but not necessary, to leave denominators in factored form.

Sample Set A

Perform the following multiplications.

Example $\PageIndex{1}$

$\dfrac{3}{4} \cdot \dfrac{1}{2} = \dfrac{3 \cdot 1 } { 4 \cdot 2} = \dfrac{3}{8}$

Example $\PageIndex{2}$

$\dfrac{8}{9} \cdot \dfrac{1}{6}=\dfrac{_\cancel{8}^{4}}{9} \cdot \dfrac{1}{^\cancel{6}_{3}}=\dfrac{4 \cdot 1}{9 \cdot 3}=\dfrac{4}{27}$

Example $\PageIndex{3}$

$\dfrac{3x}{5y} \cdot \dfrac{7}{12y} = \dfrac{_\cancel{3}^{1}x}{5y} \cdot \dfrac{7}{^\cancel{12}_{4}y} = \dfrac{x \cdot 7}{5y \cdot 4y} = \dfrac{7x}{20y^2}$

Example $\PageIndex{4}$

$\dfrac{x+4}{x-2} \cdot \dfrac{x+7}{x+4}$ Divide out the common factor $x + 4$.

$\dfrac{\cancel{x+4}}{x-2} \cdot \dfrac{x+7}{\cancel{x+4}}$ Multiply the numerators and denominators together.

$\dfrac{x+7}{x-2}$

Example $\PageIndex{5}$

$\dfrac{x^2 + x - 6}{x^2 - 4x + 3} \cdot \dfrac{x^2 - 2x - 3}{x^2 + 4x - 12}$. Factor.

$\dfrac{(x+3)(x-2)}{(x-3)(x-1)} \cdot \dfrac{(x-3)(x+1)}{(x+6)(x-2)}$. Divide out the common factors $x-2$ and $x-3$.

$\dfrac{(x+3)\cancel{(x-2)}}{\cancel{(x-3)}(x-1)} \cdot \dfrac{\cancel{(x-3)}(x+1)}{(x+6)\cancel{(x-2)}}$ Multiply.

$\dfrac{(x+3)(x+1)}{(x-1)(x+6)}$ or $\dfrac{(x62 + 4x + 3}{(x-1)(x+6)}$ or $\dfrac{(x^2 + 4x + 3}{x^2 + 5x - 6}$

Each of these three forms is an acceptable form of the same answer.

Example $\PageIndex{6}$

$\dfrac{2x+6}{6x-16} \cdot \dfrac{x^2 - 4}{x^2 - x - 12}$. Factor.

$\dfrac{2(x+3)}{8(x-2)} \cdot \dfrac{(x+2)(x-2)}{(x-4)(x+3)}$. Divide out the common factors $2, x+3$ and $x-2$.

$\dfrac{\cancel{2}\cancel{(x+3)}}{\cancel{8}\cancel{(x-2)}} \cdot \dfrac{(x+2)\cancel{(x-2)}}{\cancel{(x+3)}(x-4)}$ Multiply.

$\dfrac{x+2}{4(x-4)}$ or $\dfrac{x+2}{4x - 16}$

Both these forms are acceptable forms of the same answer.

Example $\PageIndex{7}$

$3x^2 \cdot \dfrac{x+7}{x-5}$. Rewrite $3x^2$ as $\dfrac{3x^2}{1}$.

$\dfrac{3x^2}{1} \cdot \dfrac{x+7}{x-5}$. Multiply.

$\dfrac{3x^2(x+7)}{x-5}$

Example $\PageIndex{8}$

$(x-3) \cdot \dfrac{4x-9}{x^2 - 6x + 9}$

$\dfrac{\cancel{(x-3)}}{1} \cdot \dfrac{4x-9}{\cancel{(x-3)}(x-3)}$

$\dfrac{4x-9}{x-3}$

Example $\PageIndex{9}$

$\dfrac{-x^2 - 3x - 2}{x^2 + 8x + 15} \cdot \dfrac{4x + 20}{x^2 + 2x}$. Factor $-1$ from the first numerator.

$\dfrac{-(x^2 + 3x + 2)}{x^2 + 8x + 15} \cdot \dfrac{4x + 20}{x^2 + 2}$. Factor.

$\dfrac{-(x+1)\cancel{(x+2)}}{(x+3)\cancel{(x+5)}} \cdot \dfrac{4 \cancel{(x+5)}}{x \cancel{(x+2)}}$ Multiply.

$\dfrac{-4(x + 1)}{x(x+3)} = \dfrac{-4x - 1}{x(x+3)}$ or $\dfrac{-4x - 1}{x^2 + 3x}$

Practice Set A

Perform each multiplication.

Practice Problem $\PageIndex{1}$

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

Answer

$\dfrac{10}{7}$

Practice Problem $\PageIndex{2}$

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

Answer

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

Practice Problem $\PageIndex{3}$

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

Answer

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

Practice Problem $\PageIndex{4}$

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

Answer

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

Practice Problem $\PageIndex{5}$

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

Answer

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

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

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

Sample Set B

Perform the following divisions.

Example $\PageIndex{10}$

$\dfrac{6x^2}{5a} \div \dfrac{2x}{10a^3}$ Invert the divisor and multiply

$\dfrac{_\cancel{6}^{3} x^{\not 2}}{\not 5 \not a} \cdot \dfrac{_\cancel{10}^{2} a^{_\cancel{3}^{2}}}{\not 2 \not x} = \dfrac{3x \cdot 2a^2}{1} = 6a^2x$

Example $\PageIndex{11}$

$\dfrac{x^2 + 3x - 10}{2x - 2} \div \dfrac{x^2 + 9x + 20}{x^2 + 3x - 4}$ Invert and Multiply.

$\dfrac{x^2 + 3x - 10}{2x - 2} \cdot \dfrac{x^2 + 3x - 4}{x^2 + 9x + 20}$. Factor

$\dfrac{\cancel{(x+5)}(x-2)}{2\cancel{(x-2)}} \cdot \dfrac{\cancel{(x+4)}\cancel{(x-1)}}{\cancel{(x+5)}\cancel{(x+4)}}$

$\dfrac{x-2}{2}$

Example $\PageIndex{12}$

$(4x + 7) \div \dfrac{12x + 21}{x-2}$. Write $4x + 7$ as $\dfrac{4x + 7}{1}$.

$\dfrac{4x + 7}{1} \div \dfrac{12x + 21}{x-2}$ Invert and multiply.

$\dfrac{4x + 7}{1} \div \dfrac{x-2}{12x + 21}$. Factor.

$\dfrac{\cancel{4x + 7}}{1} \cdot \dfrac{x-2}{3 \cancel{(4x+7)}} = \dfrac{x-2}{3}$

Practice Set B

Perform each division.

Practice Problem $\PageIndex{6}$

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

Answer

$20a^2mn$

Practice Problem $\PageIndex{7}$

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

Answer

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

Practice Problem $\PageIndex{8}$

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

Answer

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

Exercises

For the following problems, perform the multiplication and divisions.

Exercise $\PageIndex{1}$

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

Answer

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

Exercise $\PageIndex{2}$

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

Exercise $\PageIndex{3}$

$\dfrac{a}{b} \cdot \dfrac{b}{a}$

Answer

$1$

Exercise $\PageIndex{4}$

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

Exercise $\PageIndex{5}$

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

Answer

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

Exercise $\PageIndex{6}$

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

Exercise $\PageIndex{7}$

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

Answer

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

Exercise $\PageIndex{8}$

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

Exercise $\PageIndex{9}$

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

Answer

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

Exercise $\PageIndex{10}$

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

Exercise $\PageIndex{11}$

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

Answer

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

Exercise $\PageIndex{12}$

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

Exercise $\PageIndex{13}$

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

Answer

$-15a^2bpq$

Exercise $\PageIndex{14}$

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

Exercise $\PageIndex{15}$

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

Answer

$3a$

Exercise $\PageIndex{16}$

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

Exercise $\PageIndex{17}$

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

Answer

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

Exercise $\PageIndex{18}$

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

Exercise $\PageIndex{19}$

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

Answer

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

Exercise $\PageIndex{20}$

$\dfrac{14mn^3}{25n^6} \div \dfrac{6a^2}{15x^2}$

Answer

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

Exercise $\PageIndex{21}$

$\dfrac{24p^3q}{9mn^3} \div \dfrac{10pq}{-21n^2}$

Exercise $\PageIndex{22}$

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

Answer

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

Exercise $\PageIndex{23}$

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

Exercise $\PageIndex{24}$

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

Answer

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

Exercise $\PageIndex{25}$

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

Exercise $\PageIndex{26}$

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

Answer

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

Exercise $\PageIndex{27}$

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

Exercise $\PageIndex{28}$

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

Answer

$\dfrac{3}{4}$

Exercise $\PageIndex{29}$

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

Exercise $\PageIndex{30}$

$x^3 \cdot \dfrac{4ab}{x}$

Answer

$4abx^2$

Exercise $\PageIndex{31}$

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

Exercise $\PageIndex{32}$

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

Answer

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

Exercise $\PageIndex{33}$

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

Exercise $\PageIndex{34}$

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

Answer

$49m^3n$

Exercise $\PageIndex{35}$

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

Exercise $\PageIndex{36}$

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

Answer

$x−1$

Exercise $\PageIndex{37}$

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

Exercise $\PageIndex{38}$

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

Answer

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

Exercise $\PageIndex{39}$

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

Exercise $\PageIndex{40}$

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

Answer

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

Exercise $\PageIndex{41}$

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

Exercise $\PageIndex{42}$

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

Answer

$(y−1)$

Exercise $\PageIndex{43}$

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

Exercise $\PageIndex{44}$

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

Answer

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

Exercise $\PageIndex{45}$

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

Exercise $\PageIndex{46}$

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

Answer

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

Exercise $\PageIndex{47}$

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

Exercise $\PageIndex{48}$

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

Answer

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

Exercise $\PageIndex{49}$

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

Exercise $\PageIndex{50}$

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

Answer

$1$

Exercise $\PageIndex{51}$

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

Exercise $\PageIndex{52}$

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

Answer

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

Exercise $\PageIndex{53}$

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

Exercise $\PageIndex{54}$

$\dfrac{x^3y - x^2y^2}{x^2y - y^2} \cdot \dfrac{x^2 - y}{x - xy}$

Answer

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

Exercise $\PageIndex{55}$

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

Exercise $\PageIndex{56}$

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

Answer

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

Exercise $\PageIndex{57}$

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

Exercise $\PageIndex{58}$

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

Answer

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

Exercise $\PageIndex{59}$

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

Exercise $\PageIndex{60}$

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

Answer

$\dfrac{2ab^2}{5}$

Exercise $\PageIndex{61}$

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

Exercise $\PageIndex{62}$

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

Answer

$-6x^2y^4$

Exercise $\PageIndex{63}$

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

Exercise $\PageIndex{64}$

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

Answer

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

Exercise $\PageIndex{65}$

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

Exercise $\PageIndex{66}$

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

Answer

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

Exercise $\PageIndex{67}$

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

Exercise $\PageIndex{68}$

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

Answer

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

Exercise $\PageIndex{69}$

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

Exercise $\PageIndex{70}$

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

Answer

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

Exercise $\PageIndex{71}$

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

Exercise $\PageIndex{72}$

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

Answer

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

Exercise $\PageIndex{73}$

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

Exercise $\PageIndex{74}$

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

Answer

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

Exercise $\PageIndex{75}$

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

Exercise $\PageIndex{76}$

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

Answer

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

Exercises For Review

Exercise $\PageIndex{77}$

If $a < 0$, then $|a| =$.

Exercise $\PageIndex{78}$

Classify the polynomial $4xy+2y$ as a monomial, binomial, or trinomial. State its degree and write the numerical coefficient of each term.

Answer

binomial; 2; 4, 2

Exercise $\PageIndex{79}$

Find the product: $y^2(2y - 1)(2y + 1)$

Exercise $\PageIndex{80}$

Translate the sentence “four less than twice some number is two more than the number” into an equation.

Answer

$2x−4=x+2$

Exercise $\PageIndex{81}$

Reduce the fraction $\dfrac{x^2 - 4x + 4}{x^2 - 4}$