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Rational Expressions and Equations,
Simplifying Introduction

Rational expressions and equations occur fairly often in different branches of Math.
As such, it is important to be able to recognise such expressions and equations when you encounter them.

Rational Expression:

A rational expression is a quotient that contains two polynomials.

Thus a fraction that has a polynomial as both the numerator and the denominator.

The following are examples of rational expressions.

\frac{2x^{\tt{2}} + 3}{x^{\tt{3}} \space {\text{–}} \space 4}             \frac{5x^{\tt{3}} + x \space {\text{–}} \space 9}{2x^{\tt{2}} + 5x + 7}             \frac{x \space {\text{–}} \space 6}{x}

Even something like   \frac{2}{x^{\tt{2}}}   is a rational expression.

As a single number like  2  can be considered a type of polynomial.

In general.
Rational Expression  =  \color{blue}\frac{P(x)}{Q(x)}

Where  P(x)  and  Q(x)  are polynomials,  and  Q(x) \space \cancel{=} \space 0.

Rational Equation:

A rational equation is an equation that contains one or more rational expressions.

The following are examples of rational equations.

\frac{x}{2} \space {\scriptsize{=}} \space \frac{5}{2}       ,       \frac{x \space {\text{–}} \space 2}{7} \space {\scriptsize{=}} \space \frac{x}{3}

Solving such equations is demonstrated in the Algebra 2 section, but it’s important to mention them when introducing rational expressions.

Proper and Improper
Rational Expressions:

Just like in the case of fractions with numbers.

Rational expressions can be classed as ‘proper’ or ‘improper’ depending on the polynomials involved.

With whole number fractions, a proper fraction is where the numerator on top is smaller than the denominator below.
While an improper fraction is where the numerator is greater than the denominator.

\bf{\frac{3}{4}}   is a proper fraction.         \bf{\frac{7}{4}}   is an improper fraction.

A similar principle follows with rational expressions.
Whether a rational expressions is proper or improper depends on the degree of the polynomials, which is the largest exponent.

If the polynomial below has the larger degree, then the rational expression is ‘proper’.
If the polynomial above has the larger degree, then the rational expression is ‘improper’.

\frac{x \space + \space 1}{x^{\tt{2}} \space {\text{–}} \space 3}   is a proper rational expression.         \frac{2x^{\tt{3}} \space + \space 4}{x^{\tt{2}} \space + \space 3}   is an improper rational expression.

Rational Expressions and Equations,
Simplifying Rational Expressions

Like with number fractions, rational expressions can sometimes be simplified further from the form that they are first presented in.

What we want to do is to look to try and find common factors between the numerator and the denominator, so that we can cancel terms out to make the rational expression simpler.

Example 1

Let’s consider a basic rational expression.     \frac{a^{\tt{3}}}{4a}

Both the top and bottom can be factored to give us a common factor.

a^3 = \space a^2 \cdot a     ,     4a = \space 4 \cdot a

\frac{a^{\tt{3}}}{4a} \space {\scriptsize{=}} \space \frac{a^{\tt{2}} \cdot a}{4 \cdot a} \space {\scriptsize{=}} \space \frac{a^{\tt{2}} \cdot {\cancel{a}}}{4 \cdot {\cancel{a}}} \space {\scriptsize{=}} \space \frac{a^{\tt{2}}}{4}

The fraction  \frac{a^{\tt{2}}}{4}  is a simpler form of the original rational expression.

We couldn’t factor any further in search of common factors.

Example 2

The same logic applies to larger rational expressions also.

We could have a rational expression such as    \frac{(x + 1)(x \space {\text{–}} \space2)}{(x + 1)(x + 5)}.

The  (x + 1)‘s  can be cancelled out just like a single variable can.

\frac{(x + 1)(x \space {\text{–}} \space2)}{(x + 1)(x + 5)} \space {\scriptsize{=}} \space \frac{({\cancel{x + 1}})(x \space {\text{–}} \space2)}{({\cancel{x + 1}})(x + 5)} \space {\scriptsize{=}} \space \frac{(x \space {\text{–}} \space2)}{(x + 5)}

No further factoring can be done from here, so   \frac{(x \space {\text{–}} \space2)}{(x + 5)}   is the simplest form of the rational expression.

Domain Restrictions on
Rational Expressions

Sometimes it can be required to specify what values can and cannot be input into a rational expression.

Mostly this will be taking in to account that we can’t divide by  0.

So if there is a possibility of a value making the denominator of a rational expression  0.

We might have to eliminate that value from the possible values that can be in the domain.

Looking again at the second example from above.

\frac{(x + 1)(x \space {\text{–}} \space2)}{(x + 1)(x + 5)}

If  x  was either  -1  or  -5  the denominator below would give a result of  0.

So we could write our simplified expression as   \frac{(x \space {\text{–}} \space2)}{(x + 5)}     x \space {\cancel{=}} \space {\text{-}}1 , {\text{-}}5.

Taking into account the whole denominator before the simplified form.

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