Week 14: Multiplying and Dividing Rational Expressions

Published by on May 20, 2018 | Leave a response

The strategies for multiplying and dividing rational numbers can be used to multiply and divide rational expressions.

*Multiplying:

  • Identify the non-permissible value(s) of each expression, the value of x that makes the denominator(s) equal to 0.
  • Divide the numerators and denominators by their common factors
  • Multiply the numerators; multiply the denominators.

Example: \frac{2x^2(x+2)}{3x} . \frac{5(x-4)}{8x(x+2)}

The non-permissible values are x = 0 and x = -2

\frac{2x^2(x+2)}{3x} . \frac{5(x-4)}{8x(x+2)}

Divide the numerators and denominators by x^2, (x+2), and 2, we have

=\frac{5(x-4)}{12} with x \neq 0, -2

Example: \frac{2x^2 - x - 1}{x^2 + 2x - 3} . \frac{4x^2 + 28x + 48}{2x^2 - 13x - 7}

=\frac{(2x+1)(x-1)}{(x+3)(x-1)} . \frac{2(x+4)(x+3)}{(2x+1)(x-7)}

The non-permissible values are x = -3, x = 1, x = \frac{-1}{2}, and x=7.

Divide the numerators and the denominators by (x+3), (x-1), and (2x+1).

=\frac{4(x+4)}{x-7} with x \neq -3, 1, 7, \frac{-1}{2}

*Dividing:

  • Identify the non-permissible value(s) of each expression. These include the values that make the denominator(s) equal to 0. In addition, since division by 0 is not permitted, any value that makes the numerator of the dividor 0 is a non-permissible value.
  • Change form dividing the divisor to multiplying the reciprocal of the divisor.
  • Multiply the numerators; multiply the denominators

Example: \frac{5(x-3)}{2x} \div \frac{10(x-3)}{3x(x+5)}

The non-permissible values are x = 0, x = -5, and x = 3

=\frac{5(x-3)}{2x} . \frac{3x(x+5)}{10(x-3)}

Divide the numerators and the denominators by 5, x, (x-3)

=\frac{3(x+5)}{4} with x \neq 0, -5, 3

Example: \frac{2s^2 - 7s -4}{6s^2 - 5s - 6} \div \frac{4s^2 +4s +1}{12s + 8}

=\frac{(2s+1)(s -4 )}{(2s -3)(3s +2)} \div \frac{(2s+1)^2}{4(3s+2)}

The non-permissible values are x = \frac {3}{2}, x = \frac{-2}{3}, x = \frac {-1}{2}

=\frac{(2s+1)(s -4 )}{(2s -3)(3s +2)} . \frac{4(3s+2)}{(2s+1)^2}

Divide the numerators and the denominators by (2s +1), (3s + 2)

=\frac{4(s-4)}{(2s-3)(2s+1)} with x \neq \frac{3}{2}, \frac{-2}{3}, \frac{-1}{2}

Bonus question: Simplify \frac{1 - \frac{2}{a}}{1- \frac{4}{a^2}}

=\frac{1- \frac{2}{a}}{1 - (\frac{2}{a})^2}

=\frac{1- \frac{2}{a}}{(1- \frac{2}{a})(1+ \frac{2}{a})}

The non-permissible values are a= 0, a=2, and a=-2

Divide the numerators and the denominators by (1- \frac{2}{a})

=\frac{1}{1+ \frac{2}{a}}

=1 \div \frac{a+2}{a}

=\frac{a}{a+2} with a \neq 0, 2, -2

 

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