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12.5: Multiplication and Division of Rational Expressions

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Learning Objectives

  • Multiply rational expressions involving monomials.
  • Multiply rational expressions involving polynomials.
  • Multiply a rational expression by a polynomial.
  • Divide rational expressions involving polynomials.
  • Divide a rational expression by a polynomial.
  • Solve real-world problems involving multiplication and division of rational. expressions

Introduction

The rules for multiplying and dividing rational expressions are the same as the rules for multiplying and dividing rational numbers. Let’s start by reviewing multiplication and division of fractions. When we multiply two fractions we multiply the numerators and denominators separately

 \frac{a} {b} \cdot \frac{c} {d} = \frac{a \cdot c} {b \cdot d}

When we divide two fractions we first change the operation to multiplication. Remember that division is the reciprocal operation of multiplication or you can think that division is the same as multiplication by the reciprocal of the number.

 \frac{a} {b} \div \frac{c} {d} = \frac{a} {b} \cdot \frac{d} {c}

The problem is completed by multiplying the numerators and denominators separately  \frac{a \cdot d} {b \cdot c}.

Multiply Rational Expressions Involving Monomials

Example 1

Multiply  \frac{4} {5} \cdot \frac{15} {8}.

Solution

We follow the multiplication rule and multiply the numerators and the denominators separately.

 \frac{4} {5} \cdot \frac{15} {8} = \frac{4 \cdot 15} {5 \cdot 8} = \frac{60} {40}

Notice that the answer is not in simplest form. We can cancel a common factor of 20 from the numerator and denominator of the answer.

 \frac{60} {40} = \frac{3} {2}

We could have obtained the same answer a different way: by reducing common factors before multiplying.

 \frac{4} {5} \cdot \frac{15} {8} = \frac{4 \cdot 15} {5 \cdot 8}

We can cancel a factor of 4 from the numerator and denominator:

\frac{4}{5}\cdot \frac{15}{8}=\frac{\cancel{4}^1 \cdot 15}{5 \cdot \cancel{8}_2}

We can also cancel a factor of 5 from the numerator and denominator:

\frac{1}{5} \cdot \frac{15}{2}=\frac{1 \cdot \cancel{15}^3}{\cancel{5}_1 \cdot 2}=\frac{1 \cdot 3}{1 \cdot 2}=\frac{3}{2}

Answer The final answer is \frac{3}{2}, no matter which you you go to arrive at it.

Multiplying rational expressions follows the same procedure.

  • Cancel common factors from the numerators and denominators of the fractions.
  • Multiply the leftover factors in the numerator and denominator.

Example 2

Multiply the following  \frac{a} {16b^8} \cdot \frac{4b^3} {5a^2}.

Solution

Cancel common factors from the numerator and denominator.

\frac{\cancel{a}^1}{\cancel{16}_4 \cdot \cancel{b^8}_{b^5}} \cdot \frac{\cancel{4}^1 \cdot \cancel{b^3}^1}{\cancel{5a^2}_a}

When we multiply the left-over factors, we get

 \frac{1} {4ab^5} \ \text{Answer}

Example 3

Multiply  9x^2 \cdot \frac{4y^2} {21x^4}.

Solution

Rewrite the problem as a product of two fractions.

 \frac{9x^2} {1} \cdot \frac{4y^2} {21x^4}

Cancel common factors from the numerator and denominator

\frac{\cancel{9}^3 \cancel{x^2}} {1} \cdot \frac{4y^2}{\cancel{21}_7 \cancel{x^4}_{x^2}}

We multiply the left-over factors and get

 \frac{12y^2} {7x^2} \ \text{Answer}

Multiply Rational Expressions Involving Polynomials

When multiplying rational expressions involving polynomials, the first step involves factoring all polynomials expressions as much as we can. We then follow the same procedure as before.

Example 4

Multiply  \frac{4x + 12} {3x^2} \cdot \frac{x} {x^2 - 9}.

Solution

Factor all polynomial expression when possible.

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

Cancel common factors in the numerator and denominator of the fractions:

\frac{4 \cancel{(x+3)}}{3 \cancel{x^2}_x} \cdot \frac{\cancel{x}}{\cancel{(x+3)}(x-3)}

Multiply the left-over factors.

 \frac{4} {3x(x - 3)} = \frac{4} {3x^2 - 9x} \ \text{Answer}

Example 5

Multiply  \frac{12x^2 - x - 6} {x^2 - 1} \cdot \frac{x^2 + 7x + 6} {4x^2 - 27x + 18}.

Solution

Factor all polynomial expression when possible.

 \frac{(3x + 2)(4x - 3)} {(x + 1) (x - 1)} \cdot \frac{(x + 1)(x + 6)} {(4x - 3) (x - 6)}

Cancel common factors in the numerator and denominator of the fractions.

\frac{(3x+2)\cancel{(4x-3)}}{\cancel{(x+1)}(x-1)}\cdot \frac{\cancel{(x+1)}(x+6)}{\cancel{(4x-3)}(x-6)}

Multiply the remaining factors.

 \frac{(3x + 2) (x + 6)} {(x - 1) (x -6)} = \frac{3x^2 + 20x + 12} {x^2 - 7x + 6} \ \text{Answer}

Multiply a Rational Expression by a Polynomial

When we multiply a rational expression by a whole number or a polynomial, we must remember that we can write the whole number (or polynomial) as a fraction with denominator equal to one. We then proceed the same way as in the previous examples.

Example 6

Multiply  \frac{3x + 18} {4x^2 + 19x - 5} \cdot x^2 + 3x - 10.

Solution

Rewrite the expression as a product of fractions.

 \frac{3x + 18} {4x^ + 19x - 5} \cdot \frac{x^2 + 3x - 10} {1}

Factor all polynomials possible and cancel common factors.

\frac{3x(x+6)}{\cancel{(x+5)}(4x-1)}\cdot \frac{(x-2)\cancel{(x+5)}}{1}

Multiply the remaining factors.

 \frac{(3x + 18)(x - 2)} {4x - 1} = \frac{3x^2 + 12x - 36} {4x - 1}

Divide Rational Expressions Involving Polynomials

Since division is the reciprocal of the multiplication operation, we first rewrite the division problem as a multiplication problem and then proceed with the multiplication as outlined in the previous example.

Note: Remember that  \frac{a} {b} \div \frac{c} {d} = \frac{a} {b} \cdot \frac{d} {c}. The first fraction remains the same and you take the reciprical of the second fraction. Do not fall in the common trap of flipping the first fraction.

Example 7

Divide  \frac{4x^2} {15} \div \frac{6x} {5}.

Solution

First convert into a multiplication problem by flipping what we are dividing by and then simplify as usual.

\frac{\cancel{4}^2 \cancel{x^2}^x}{\cancel{15}}\cdot \frac{\cancel{5}^2}{\cancel{6x}_1}=\frac{2\cdot x\cdot 1}{3\cdot 3\cdot 1}=\frac{2x}{9}

Example 8

Divide  \frac{3x^2 - 15x} {2x^2 + 3x - 14} \div \frac{x^2 - 25} {2x^2 + 13x + 21}.

Solution

First convert into a multiplication problem by flipping what we are dividing by and then simplify as usual.

 \frac{3x^2 - 15x} {2x^2 + 3x - 14} \cdot \frac{2x^2 + 13x + 21} {x^2 - 25}

Factor all polynomials and cancel common factors.

\frac{3x\cancel{(x-5)}}{\cancel{(2x+7)}(x-2)}\cdot \frac{\cancel{(2x+7)}(x-2)}{\cancel{(x-5)}(x+5)}

Multiply the remaining factors.

 \frac{3x(x + 3)} {(x - 2)(x + 5)} = \frac{3x^2 + 9x} {x^2 + 3x - 10}\ \text{Answer}.

Divide a Rational Expression by a Polynomial

When we divide a rational expression by a whole number or a polynomial, we must remember that we can write the whole number (or polynomial) as a fraction with denominator equal to one. We then proceed the same way as in the previous examples.

Example 9

Divide  \frac{9x^2 - 4} {2x - 2} \div {21x^2 - 2x - 8}.

Solution

Rewrite the expression as a division of fractions.

 \frac{9x^2 - 4} {2x - 2} \div \frac{21x^2 - 2x - 8} {1}

Convert into a multiplication problem by taking the reciprocal of the divisor (i.e. “what we are dividing by”).

 \frac{9x^2 - 4} {2x - 2} \cdot \frac{1} {21x^2 - 2x - 8}

Factor all polynomials and cancel common factors.

\frac{\cancel{(3x-2)}(3x+2)}{2(x-1)}\cdot \frac{1}{\cancel{(3x-2)}(7x+4)}

Multiply the remaining factors.

 \frac{3x + 2} {14x^2 - 6x - 8}.

Solve Real-World Problems Involving Multiplication and Division of Rational Expressions

Example 10

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 x^{3} - 9x, where x is the number of bowls of cereal he had for breakfast (1 \le x \le 6). Marciel’s training distance (in miles), if he eats x bowls of cereal, is 3x^{2} - 9x. 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?

Solution

 \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}

If x = 5, then

\text{time} = \frac{3} {5 + 3} = \frac{3} {8}

Answer Marciel will run for \frac{3}{8} of an hour.

Review Questions

Perform the indicated operation and reduce the answer to lowest terms

  1.  \frac{x^3} {2y^3} \cdot \frac{2y^2} {x}
  2.  2xy \div \frac{2x^2} {y}
  3.  \frac{2x} {y^2} \cdot \frac{4y} {5x}
  4.  2xy \cdot \frac{2y^2} {x^3}
  5.  \frac{4y^2 - 1} {y^2 - 9} \cdot \frac{y - 3} {2y - 1}
  6.  \frac{6ab} {a^2} \cdot \frac{a^3b} {3b^2}
  7.  \frac{x^2} {x - 1} \div \frac{x} {x^2 + x -2}
  8.  \frac{33a^2} {-5} \cdot \frac{20} {11a^3}
  9.  \frac{a^2 + 2ab + b^2} {ab^2 - a^2b} \div (a + b)
  10.  \frac{2x^2 + 2x - 24} {x^2 + 3x} \cdot \frac{x^2 + x - 6} {x + 4}
  11.  \frac{3 - x} {3x - 5} \div \frac{x^2 - 9} {2x^2 - 8x - 10}
  12.  \frac{x^2 - 25} {x + 3} \div (x - 5)
  13.  \frac{2x + 1} {2x - 1} \div \frac{4x^2 - 1} {1 - 2x}
  14.  \frac{x} {x - 5} \cdot \frac{x^2 - 8x + 15} {x^2 - 3x}
  15.  \frac{3x^2 + 5x - 12} {x^2 - 9} \div \frac{3x - 4} {3x+ 4}
  16.  \frac{5x^2 + 16x + 3} {36x^2 - 25} \cdot (6x^2 + 5x)
  17.  \frac{x^2 + 7x + 10} {x^2 - 9} \cdot \frac{x^2 - 3x} {3x^2 + 4x - 4}
  18.  \frac{x^2 + x - 12} {x^2 + 4x + 4} \div \frac{x - 3} {x + 2}
  19.  \frac{x^4 - 16} {x^2 - 9} \div \frac{x^2 + 4} {x^2 + 6x + 9}
  20.  \frac{x^2 + 8x + 16} {7x^2 + 9x + 2} \div \frac{7x + 2} {x^2 + 4x}
  21. Maria’s recipe asks for 2 \frac{1} {2} times more flour than sugar. How many cups of flour should she mix in if she uses 3 \frac{1} {3} cups of sugar?
  22. 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?
  23. Ohm’s Law states that in an electrical circuit I = \frac{V} {R_c}. The total resistance for resistors placed in parallel is given by  \frac{1} {R_{tot}} = \frac{1} {R_1} + \frac{1} {R_2}. Write the formula for the electric current in term of the component resistances: R_1 and R_2.

Review Answers

  1.  \frac{x^2} {y}
  2.  \frac{y^2} {x}
  3.  \frac{8} {5y}
  4.  \frac{47^3} {x^2}
  5.  \frac{2y + 1} {y + 3}
  6. 2a^2
  7. x^2 + 2x
  8.  \frac{-12} {a}
  9.  \frac{a + b} {ab^2 - a^2b}
  10.  \frac{2x^2 - 10x + 12} {x}
  11.  \frac{-2x^2 + 8x + 10} {3x^2 + 4x - 15}
  12.  \frac{x + 5} {x + 3}
  13.  \frac{1} {1 - 2x}
  14. 1
  15.  \frac{3x + 4} {x - 3}
  16.  \frac{5x^3 + 16x^2 + 3x} {(6x - 5)}
  17.  \frac{x^2 + 5x} {3x^2 - 11x + 6}
  18.  \frac{x + 4} {x + 2}
  19.  \frac{x^2 - 4} {x^2 - 9}
  20.  \frac{x + 4} {x^2 + x}
  21.  8\frac{1} {3} cups
  22.  \frac{s} {s + 15}
  23.  I = \frac{E} {R_1} + \frac{E} {R_2}

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