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4. Rational Functions & Expressions
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United States of AmericaFL
Grade 912

4.03 Multiplying and dividing rational expressions

Lesson
Practice
Lesson

Concept summary

To multiply two (or more) rational expressions together, we multiply the numerators to form the new numerator and multiply the denominators to form the new denominator - the same process as for multiplying fractions:\frac{A}{B} \cdot \frac{C}{D} = \frac{AC}{BD}

Common factors, in particular the greatest common factor (GCF), between any of the numerators and denominators can be divided out to simplify the expression. This can be done before or after evaluating the multiplication.

To divide two rational expressions, we multiply the first rational expression by the reciprocal of the second rational expression - the same process as for dividing fractions:\frac{A}{B} \div \frac{C}{D} = \frac{A}{B} \cdot \frac{D}{C} = \frac{AD}{BC}

Worked examples

Example 1

Fully simplify the expression\frac{21a^3 b^6}{20 c^2} \cdot \frac{16 a^2 c^3}{15 b^6}

Approach

It will be easier to identify any common factors between numerators and denominators first and divide them out before multiplying. We can start by splitting the coefficients into factors.

Solution

\displaystyle \frac{21a^3 b^6}{20 c^2} \cdot \frac{16 a^2 c^3}{15 b^6}\displaystyle =\displaystyle \frac{7 \cdot 3 \cdot a^3 b^6}{5 \cdot 4 \cdot c^2} \cdot \frac{4 \cdot 4 \cdot a^2 c^3}{5 \cdot 3 \cdot b^6}Write coefficients as products of factors
\displaystyle =\displaystyle \frac{7 a^3}{5} \cdot \frac{4 a^2 c}{5}Divide out common factors between numerators and denominators
\displaystyle =\displaystyle \frac{28 a^5 c}{25}Multiply across numerators and denominators

Reflection

While it is generally easier to divide out common factors first, we can achieve the same result by performing the multiplication first and then looking for common factors at the end:

\displaystyle \frac{21a^3 b^6}{20 c^2} \cdot \frac{16 a^2 c^3}{15 b^6}\displaystyle =\displaystyle \frac{336a^5 b^6 c^3}{300 b^6 c^2}Multiply across numerators and denominators
\displaystyle =\displaystyle \frac{112a^5 c}{100}Divide out common factor of 3b^6 c^2
\displaystyle =\displaystyle \frac{28a^5 c}{25}Divide out remaining common factor of 4

Doing the steps in this order usually results in much larger numbers to deal with. Dividing out common factors can be done across multiple steps, however, looking for more common factors after each step.

Example 2

Fully simplify the expression\frac{10 x}{y^2 z} \div \frac{3 x^2 z}{10 y}

Approach

We can first turn the division into a multiplication by taking the reciprocal of the second fraction. We then again want to identify common factors that can be divided out before performing the multiplication.

Solution

\displaystyle \frac{10 x}{y^2 z} \div \frac{3 x^2 z}{10 y}\displaystyle =\displaystyle \frac{10 x}{y^2 z} \cdot \frac{10 y}{3 x^2 z}Rewrite division as multiplication
\displaystyle =\displaystyle \frac{10}{y z} \cdot \frac{10}{3 x z}Divide out common factors between numerators and denominators
\displaystyle =\displaystyle \frac{100}{3 x y z^2}Multiply across numerators and denominators

Reflection

Note that the first step was to rewrite the division as a multiplication, and we could then treat the expression as normal for a multiplication problem.

While there are some simplification steps that can be done in division form, it is much easier to make a mistake when doing so.

Example 3

Fully simplify the expression\frac{x^2 + 3x - 10}{x^3 - 8} \cdot \frac{x^2 - 9}{x^2 + 2x - 15}

Approach

To find any simplification here, we want to rewrite each numerator and denominator in terms of its factors. This will be easier to do before multiplying, while the expressions are of lower degree.

In particular, notice that three of the expressions are quadratic, and the other one is a difference of two cubes, so we have factoring techniques we can use on each part of the expression.

Solution

\displaystyle \frac{x^2 + 3x - 10}{x^3 - 8} \cdot \frac{x^2 - 9}{x^2 + 2x - 15}\displaystyle =\displaystyle \frac{\left(x + 5\right)\left(x - 2\right)}{\left(x - 2\right)\left(x^2 + 2x + 4\right)} \cdot \frac{\left(x - 3\right)\left(x + 3\right)}{\left(x + 5\right)\left(x - 3\right)}Factor each expression
\displaystyle =\displaystyle \frac{1}{x^2 + 2x + 4} \cdot \frac{x + 3}{1}Divide out common factors
\displaystyle =\displaystyle \frac{x + 3}{x^2 + 2x + 4}Simplify the product

Reflection

Notice that the factors of \left(x - 2\right) and \left(x - 3\right) were divided out within a fraction, while the factors of \left(x + 5\right) were divided out across fractions.

Both types of simplification are fine, as long as one factor appears in a numerator and the other appears in a denominator.

Outcomes

MA.912.AR.1.8

Rewrite a polynomial expression as a product of polynomials over the real or complex number system.

MA.912.AR.1.9

Apply previous understanding of rational number operations to add, subtract, multiply and divide rational algebraic expressions.

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