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Division of Rational Expressions

Invert, cancel, multiply, and reduce fractions with variables in the denominator

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Division of Rational Expressions

Suppose that the distance traveled by a hot air balloon in miles can be represented by \begin{align*}8x^3-8x,\end{align*}8x38x, while the speed of the hot air balloon in miles per hour can be represented by \begin{align*}x^2+x\end{align*}x2+x. Would you be able to find an expression for the time it takes for the hot air balloon to cover a certain distance? Could you evaluate the expression that you found for \begin{align*}x=2\end{align*}x=2

Dividing Rational Expressions

Division of rational expressions works in the same manner as multiplication. A reminder of how to divide fractions is below.

For any rational expressions \begin{align*}a \neq 0, b \neq 0, c \neq 0, d \neq 0\end{align*}a0,b0,c0,d0,

\begin{align*}\frac{a}{b} \div \frac{c}{d} \rightarrow \frac{a}{b} \cdot \frac{d}{c}=\frac{ad}{bc}\end{align*}ab÷cdabdc=adbc

Simplify the following division problems:

  1. \begin{align*}\frac{9x^2-4}{2x-2} \div \frac{21x^2-2x-8}{1}\end{align*}9x242x2÷21x22x81

 

\begin{align*}\frac{9x^2-4}{2x-2} \div \frac{21x^2-2x-8}{1} \rightarrow \frac{9x^2-4}{2x-2} \cdot \frac{1}{21x^2-2x-8}\end{align*}9x242x2÷21x22x819x242x2121x22x8

Repeat the process for multiplying rational expressions.

\begin{align*}\frac{9x^2-4}{2x-2} \cdot \frac{1}{21x^2-2x-8} & \rightarrow \frac{(3x-2)\cancel{(3x-2)}}{2(x-1)} \cdot \frac{1}{\cancel{(3x-2)}(7x+4)}\\ \frac{9x^2-4}{2x-2} \div \frac{21x^2-2x-8}{1} &= \frac{3x-2}{14x^2-6x-8}\end{align*}9x242x2121x22x89x242x2÷21x22x81(3x2)(3x2)2(x1)1(3x2)(7x+4)=3x214x26x8

  1.  \begin{align*}\frac{x^2+3x-10}{5x+15} \div \frac{x-2}{x^2+2x-3}\end{align*}

\begin{align*}\frac{x^2+3x-10}{5x+15} \div \frac{x-2}{x^2+2x-3} \rightarrow \frac{x^2+3x-10}{5x+15} \cdot \frac{x^2+2x-3}{x-2}\end{align*}

Repeat the process for multiplying rational expressions.

\begin{align*}\frac{x^2+3x-10}{5x+15} \cdot \frac{x^2+2x-3}{x-2} & \rightarrow \frac{(x+5)(x-2)}{5(x+3)} \cdot \frac{x-2}{(x+3)(x-1)}\\ \frac{(x+5)\cancel{(x-2)}}{5\cancel{(x+3)}} \cdot \frac{\cancel{x-2}}{(\cancel{x+3})(x-1)}&=\frac{x+5}{5} \cdot \frac{1}{(x-1)}=\frac{x+5}{5x-5}\\ \frac{x^2+3x-10}{5x+15} \div \frac{x-2}{x^2+2x-3} &= \frac{x+5}{5x-5}\end{align*}

Consider the following real-world application: 

Suppose Marciel is training for a running race. Marciel’s speed (in miles per hour) of his training run each morning is given by the function \begin{align*}x^3-9x\end{align*}, where \begin{align*}x\end{align*} is the number of bowls of cereal he had for breakfast \begin{align*}(1 \le x \le 6).\end{align*} Marciel’s training distance (in miles), if he eats \begin{align*}x\end{align*} bowls of cereal, is \begin{align*}3x^2-9x\end{align*}. What is the function for Marciel’s time and how long does it take Marciel to do his training run if he eats five bowls of cereal on Tuesday morning?

\begin{align*}\text{time} &= \frac{\text{distance}}{\text{speed}}\\ \text{time} &= \frac{3x^2-9x}{x^3-9x}=\frac{3x(x-3)}{x(x^2-9)}=\frac{3x\cancel{(x-3)}}{x(x+3)\cancel{(x-3)}}\\ \text{time} &= \frac{3}{x+3}\\ \text{If} \ x &= 5, \text{then}\\ \text{time} &= \frac{3}{5+3}=\frac{3}{8}\end{align*}

Marciel will run for \begin{align*}\frac{3}{8}\end{align*} of an hour.

 

Examples

Example 1

Earlier, you were told that the distance traveled by a hot air balloon in miles can be represented by \begin{align*}8x^3-8x\end{align*}, while the speed of the hot air balloon in miles per hour can be represented by \begin{align*}x^2+x\end{align*}. What expression would represent the time it takes to cover a distance? What is the time it takes the balloon to travel a distance when \begin{align*}x=2\end{align*}?

We can determine the expression for time by dividing distance by speed. 

\begin{align*}\text{time} &= \frac{\text{distance}}{\text{speed}}\\ \text{time} &= \frac{8x^3-8x}{x^2-x}=\frac{8x^2(x-1)}{x(x+1)}\\ \text{time} &= 8x-8 \end{align*}

The expression for the time it takes the balloon to cover a given distance is \begin{align*}8x-8\end{align*}.

Now, solve for \begin{align*}x=2\end{align*}

\begin{align*}\text{time} &= 8(2)-8\\ \text{time} &= 16-8\\ \text{time} &= 8\ \mathrm{hours}\end{align*}  

When \begin{align*}x=2\end{align*}, the balloon takes 8 hours to travel the given distance.

Example 2 

Simplify \begin{align*} \frac{1}{5x^2-30x+40} \div \frac{3x-6}{2x^2-8x}\end{align*}.

 

\begin{align*} \frac{1}{5x^2-30x+40} \div \frac{3x-6}{2x^2-8x} &= \frac{1}{5x^2-30x+40} \cdot \frac{2x^2-8x}{3x-6}\\ &= \frac{1}{5(x-2)(x-4)} \cdot \frac{2x(x-4)}{3(x-2)}\\ &= \frac{1}{5(x-2) \cancel{(x-4)}} \cdot \frac{2x \cancel{(x-4)}}{3(x-2)}\\ &=\frac{2x}{5(x-2)^2} \end{align*}

Review

In 1–10, perform the indicated operation and reduce the answer to lowest terms.

  1. \begin{align*}2xy \div \frac{2x^2}{y}\end{align*}
  2. \begin{align*}\frac{x^2}{x-1} \div \frac{x}{x^2+x-2}\end{align*}
  3. \begin{align*}\frac{a^2+2ab+b^2}{ab^2-a^2b} \div (a+b)\end{align*}
  4. \begin{align*}\frac{3-x}{3x-5} \div \frac{x^2-9}{2x^2-8x-10}\end{align*}
  5. \begin{align*}\frac{x^2-25}{x+3} \div (x-5)\end{align*}
  6. \begin{align*}\frac{2x+1}{2x-1} \div \frac{4x^2-1}{1-2x}\end{align*}
  7. \begin{align*}\frac{3x^2+5x-12}{x^2-9} \div \frac{3x-4}{3x+4}\end{align*}
  8. \begin{align*}\frac{x^2+x-12}{x^2+4x+4} \div \frac{x-3}{x+2}\end{align*}
  9. \begin{align*}\frac{x^4-16}{x^2-9} \div \frac{x^2+4}{x^2+6x+9}\end{align*}
  10. \begin{align*}\frac{x^2+8x+16}{7x^2+9x+2} \div \frac{7x+2}{x^2+4x}\end{align*}
  11. Maria’s recipe asks for \begin{align*}2 \frac{1}{2} \ \text{times}\end{align*} as much flour as sugar. How many cups of flour should she mix in if she uses \begin{align*}3 \frac{1}{3} \ \text{cups}\end{align*} of sugar?
  12. George drives from San Diego to Los Angeles. On the return trip, he increases his driving speed by 15 miles per hour. In terms of his initial speed, by what factor is the driving time decreased on the return trip?
  13. Ohm’s Law states that in an electrical circuit \begin{align*}I=\frac{V}{R_{tot}}.\end{align*} The total resistance for resistors placed in parallel is given by \begin{align*}\frac{1}{R_{tot}}=\frac{1}{R_1}+\frac{1}{R_2}.\end{align*} Write the formula for the electric current in terms of the component resistances: \begin{align*}R_1\end{align*} and \begin{align*}R_2\end{align*}.

Review (Answers)

To see the Review answers, open this PDF file and look for section 12.6. 

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Vocabulary

rational function

A ratio of two polynomials (a polynomial divided by another polynomial). The formal definition is: f(x)=\frac{g(x)}{h(x)}, \text{where} \ h(x) \neq 0.

reciprocal

The reciprocal of a nonzero rational expression \frac{a}{b} is \frac{b}{a}.

Least Common Denominator

The least common denominator or lowest common denominator of two fractions is the smallest number that is a multiple of both of the original denominators.

Least Common Multiple

The least common multiple of two numbers is the smallest number that is a multiple of both of the original numbers.

Rational Expression

A rational expression is a fraction with polynomials in the numerator and the denominator.

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