SAT Practice Question
\(\large\frac{5x}{6y}{\cdot}\frac{3}{10y}\) Which expression is equivalent to the above product for all \(y>0\)? a) \(\large\frac{x}{2}\) b) \(\large\frac{25x}{9}\) c) \(\large\frac{x}{2y^2}\) d) \(\large\frac{x}{4y^2}\) Source: Khan Academy |
This target addresses operations with rational expressions, including multiplication, division, addition and subtraction. Of course, we will address simplifying rational expressions, too!
A rational expression is a fraction with a polynomial in the numerator and/or the denominator.
To perform operations on rational expressions, you need to understand operations with fractions--how to simplify fractions to lowest terms, how to multiply and divide fractions, how to find least common denominators of fractions, and how to add and subtract fractions.
Multiplication and Division of Rational Expressions
To simplify fractions to lowest terms, common factors are divided out of both the numerator and the denominator. For example, \(\dfrac{15}{20}=\dfrac{5\cdot3}{5\cdot4}=\dfrac{3}{4}\) because the \(5\) will divide out of both the numerator and the denominator.
When simplifying rational expressions, the numerator and the denominator need to be factored completely, and then common factors are divided out (or you may need to apply the division of like bases rule of exponents).
Example 1: Rewrite the rational expression \(\Large{\frac{x^3-2x^2}{x^3-4x}}\) as an equivalent expression in simplest form.
A rational expression is a fraction with a polynomial in the numerator and/or the denominator.
To perform operations on rational expressions, you need to understand operations with fractions--how to simplify fractions to lowest terms, how to multiply and divide fractions, how to find least common denominators of fractions, and how to add and subtract fractions.
Multiplication and Division of Rational Expressions
To simplify fractions to lowest terms, common factors are divided out of both the numerator and the denominator. For example, \(\dfrac{15}{20}=\dfrac{5\cdot3}{5\cdot4}=\dfrac{3}{4}\) because the \(5\) will divide out of both the numerator and the denominator.
When simplifying rational expressions, the numerator and the denominator need to be factored completely, and then common factors are divided out (or you may need to apply the division of like bases rule of exponents).
Example 1: Rewrite the rational expression \(\Large{\frac{x^3-2x^2}{x^3-4x}}\) as an equivalent expression in simplest form.
\(\begin{align}&{\frac{x^2\left(x-2\right)}{x\left(x^2-4\right)}}\ & \ &\text{1) Factor the GCF from the numerator and denominator.}\\
&{\frac{x^2\left(x-2\right)}{x\left(x-2\right)\left(x+2\right)}}\ & \ &\text{2) Factor the binomial in the denominator.}\\
&{\frac{x}{\left(x+2\right)}}\ & \ &\text{3) Divide out the common factor}\ \left(x-2\right)\ \text{and apply the rule of exponents to reduce the monomial.}\\
&{\frac{x}{\left(x+2\right)}}\ & \ &\text{4) Simplify.}\end{align}\)
&{\frac{x^2\left(x-2\right)}{x\left(x-2\right)\left(x+2\right)}}\ & \ &\text{2) Factor the binomial in the denominator.}\\
&{\frac{x}{\left(x+2\right)}}\ & \ &\text{3) Divide out the common factor}\ \left(x-2\right)\ \text{and apply the rule of exponents to reduce the monomial.}\\
&{\frac{x}{\left(x+2\right)}}\ & \ &\text{4) Simplify.}\end{align}\)
To Multiply or Divide Rational Expressions:
Note: When you divide fractions, multiply the first rational expression by the reciprocal of the second rational expression.
It is important to note that only common factors can divide out. DO NOT divide out terms joined by addition or subtraction from the numerator and the denominator.
Example 2: Multiply the rational expressions \(\Large{\frac{x^2+4x+3}{x^2+5x+6}\cdot\frac{x^2-3x-10}{x^2+x}}\).
- Factor each numerator and each denominator completely.
- Divide out common factors between numerators and denominators.
- Leave the numerator and denominator in factored form.
Note: When you divide fractions, multiply the first rational expression by the reciprocal of the second rational expression.
It is important to note that only common factors can divide out. DO NOT divide out terms joined by addition or subtraction from the numerator and the denominator.
Example 2: Multiply the rational expressions \(\Large{\frac{x^2+4x+3}{x^2+5x+6}\cdot\frac{x^2-3x-10}{x^2+x}}\).
\(\begin{align}&{\frac{\left(x+3\right)\left(x+1\right)}{\left(x+3\right)\left(x+2\right)}\cdot\frac{\left(x-5\right)\left(x+2\right)}{x\left(x+1\right)}}\ & \ &\text{1) Factor each numerator and denominator.}\\
&{\frac{\left(x+3\right)\left(x+1\right)}{\left(x+3\right)\left(x+2\right)}\cdot\frac{\left(x-5\right)\left(x+2\right)}{x\left(x+1\right)}}\ & \ &\text{2) Divide out common factors (x+1, x+3 and x+2).}\\
&{\frac{\left(x-5\right)}{x}}\ & \ &\text{3) Simplify.}\end{align}\)
&{\frac{\left(x+3\right)\left(x+1\right)}{\left(x+3\right)\left(x+2\right)}\cdot\frac{\left(x-5\right)\left(x+2\right)}{x\left(x+1\right)}}\ & \ &\text{2) Divide out common factors (x+1, x+3 and x+2).}\\
&{\frac{\left(x-5\right)}{x}}\ & \ &\text{3) Simplify.}\end{align}\)
Example 3: Multiply the rational expressions \(\Large{\frac{x^2-6x+5}{x^2+2x+1}\cdot\frac{x^2-3x-4}{x^2-9x+20}}\).
Example 4: Divide the rational expressions \(\Large{\frac{x^2-4x-21}{5x+15}\div\frac{x^2+3x-70}{x^2-100}}\).
\(\begin{align}&{\frac{x^2-4x-21}{5x+15}\cdot\frac{x^2-100}{x^2+3x-70}}\ & \ &\text{1) Multiply by the reciprocal of the second rational expression.}\\
&{\frac{\left(x-7\right)\left(x+3\right)}{5\left(x+3\right)}\cdot\frac{\left(x-10\right)\left(x+10\right)}{\left(x+10\right)\left(x-7\right)}}\ & \ &\text{2) Factor each numerator and denominator.}\\
&{\frac{1}{5}\cdot\frac{\left(x-10\right)}{1}}\ & \ &\text{3) Divide out common factors (x+3, x+10 and x-7).}\\
&{\frac{x-10}{5}}\ & \ &\text{4) Simplify.}\end{align}\)
&{\frac{\left(x-7\right)\left(x+3\right)}{5\left(x+3\right)}\cdot\frac{\left(x-10\right)\left(x+10\right)}{\left(x+10\right)\left(x-7\right)}}\ & \ &\text{2) Factor each numerator and denominator.}\\
&{\frac{1}{5}\cdot\frac{\left(x-10\right)}{1}}\ & \ &\text{3) Divide out common factors (x+3, x+10 and x-7).}\\
&{\frac{x-10}{5}}\ & \ &\text{4) Simplify.}\end{align}\)
Example 5: Divide the rational expressions \(\Large{\frac{x^2-25}{4x^3-10x^2-6x}\div\frac{3x^2-14x-5}{6x^2-18x}}\).