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Polynomials and Rational Functions
Rational Functions
Algebra Textbooks Boundless Algebra Polynomials and Rational Functions Rational Functions
Algebra Textbooks Boundless Algebra Polynomials and Rational Functions
Algebra Textbooks Boundless Algebra
Algebra Textbooks
Algebra
Concept Version 14
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Simplifying, Multiplying, and Dividing Rational Expressions

A rational expression can be treated like a fraction, and can be manipulated via multiplication and division.

Learning Objective

  • Write rational expressions in lowest terms by simplifying them, using the same rules as for fractions


Key Points

    • A rational expression is a quotient of two polynomials, where the polynomial in the denominator is not zero.
    • Rational expressions can often be simplified by removing terms that can be factored out of the numerator and denominator. These can be either numbers or functions of $x$.
    • Rational expressions can be multiplied together. The numerators of each are multiplied together, as well as their denominators. Sometimes, it is possible to simplify the resulting fraction.
    • Rational expressions can be divided by one another. This follows the rules for dividing fractions, where the dividend is multiplied by the reciprocal of the divisor. 

Terms

  • expression

    An arrangement of symbols denoting values, operations performed on them, and grouping symbols, e.g. $\displaystyle \frac{(2x+4)}{2}$

  • rational expression

    An expression that can be expressed as the quotient of two polynomials, where the polynomial in the denominator is not zero.

  • polynomial

    An expression consisting of a sum of a finite number of terms, each term being the product of a constant coefficient and one or more variables raised to a non-negative integer power, such as $a_n x^n + a_{n-1}x^{n-1} + ... + a_0 x^0$. Importantly, because all exponents are positive, it is impossible to divide by $x$.


Full Text

A rational expression is a fraction involving polynomials, where the polynomial in the denominator is not zero. Just like a fraction involving numbers, a rational expression can be simplified, multiplied, and divided. The rules for performing these operations often mirror the rules for simplifying, multiplying, and dividing fractions. Performing these operations on rational expressions often involves factoring polynomial expressions out of the numerator and denominator.

Simplifying a Rational Expression

Rational expressions can be simplified by factoring the numerator and denominator where possible, and canceling terms. 

As a first example, consider the rational expression $\frac { 3x^3 }{ x }$. This can be simplified by canceling out one factor of $x$ in the numerator and denominator, which gives the expression $3x^2$.

Note that the domain of the equation $f(x) = \frac{3x^3}{x}$ does not include $x=0$, as this would cause division by $0$. The latter form is a simplified version of the former graphically.

Consider a more complicated example:

$\displaystyle \frac { x^2+5x+6 }{ 2x^2+5x+2 }$

This expression must first be factored to provide the expression

$\displaystyle \frac {(x+2)(x+3)}{(2x+1)(x+2)}$

which, after canceling the common factor of $(x+2)$ from both the numerator and denominator, gives the simplified expression

$\displaystyle \frac {x+3}{2x+1}$

Multiplying Rational Expressions

Rational expressions can be multiplied and divided in a similar manner to fractions. Recall that when two fractions are multiplied together, their numerators are multiplied to yield the numerator of their product, and their denominators are multiplied to yield the denominator of their product.

For a simple example, consider the following, where a rational expression is multiplied by a fraction of whole numbers:

$\displaystyle \frac {x^2+3}{2x-3} \times \frac{2}{3}$

Following the rule for multiplying fractions, simply multiply their respective numerators and denominators: 

$\displaystyle \frac {2(x^2+3)}{3(2x-3)}$

This can be multiplied through to yield $\displaystyle \frac {2x^2+6}{6x-9}$

Notice that we multiplied the numerators together and the denominators together, but we did not multiply the numerator by the denominator or vice-versa.

We follow the same rules to multiply two rational expressions together. The operations are slightly more complicated, as there may be a need to simplify the resulting expression.

Example 1

Consider the following:

$\displaystyle \frac {x+1}{x-1} \times \frac {x+2}{x+3}$  

Multiplying these two expressions, we have the product:

 $\displaystyle\frac {(x+1)(x+2)}{(x-1)(x+3)}$

Multiplying out the numerator and denominator, this can be written as:

$\displaystyle \frac {x^2+3x+2}{x^2+2x-3}$ 

Notice that this expression cannot be simplified further.

Dividing Rational Expressions

Dividing rational expressions follows the same rules as dividing fractions. Recall the rule for dividing fractions: the dividend is multiplied by the reciprocal of the divisor. The same applies to dividing rational expressions; the first expression is multiplied by the reciprocal of the second.

Example 2

Consider the following:

 $\displaystyle \frac {x+1}{x-1} \div \frac {x+2}{x+3}$

Rather than divide the expressions, we multiply $\displaystyle \frac {x+1}{x-1}$ by the reciprocal of $\displaystyle \frac {x+2}{x+3}$:

$\displaystyle \frac{x+1}{x-1} \times \frac {x+3}{x+2}$

Then, multiplication is carried out in the same way as described above:

$\displaystyle \frac{(x+1)(x+3)}{(x-1)(x+2)} = \frac{x^2 + 3x +3}{x^2 + x - 2}$

The expression cannot be simplified further.

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