8.1 Simplify rational expressions  (Page 2/5)

Evaluate rational expressions

To evaluate a rational expression, we substitute values of the variables into the expression and simplify, just as we have for many other expressions in this book.

Evaluate $\frac{2x+3}{3x-5}$ for each value:

ⓐ $x=0$ ⓑ $x=2$ ⓒ $x=\mathrm{-3}$

Solution

ⓐ

Simplify.

ⓑ

Simplify.

ⓒ

Simplify.

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Evaluate $\frac{x^2+8x+7}{x^2-4}$ for each value:

ⓐ $x=0$ ⓑ $x=2$ ⓒ $x=\mathrm{-1}$

Solution

ⓐ

Simplify.

ⓑ

Simplify.
This rational expression is undefined for x = 2.

ⓒ

Simplify.

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Remember that a fraction is simplified when it has no common factors, other than 1, in its numerator and denominator. When we evaluate a rational expression, we make sure to simplify the resulting fraction.

Evaluate $\frac{a^2+2ab+b^2}{3a{b}^3}$ for each value:

ⓐ $a=1,b=2$ ⓑ $a=\mathrm{-2},b=\mathrm{-1}$ ⓒ $a=\frac{1}{3},b=0$

Solution

ⓐ

	$\frac{a^2+2ab+b^2}{3a{b}^2}$  when  $a=1,b=2$ .
Simplify.

ⓑ

	$\frac{a^2+2ab+b^2}{3a{b}^2}$  when  $a=\mathrm{-2},b=\mathrm{-1}$ .
Simplify.

ⓒ

	$\frac{a^2+2ab+b^2}{3a{b}^2}$  when  $a=\frac{1}{3},b=0$ .
Simplify.
	The expression is undefined.

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Simplify rational expressions

Just like a fraction is considered simplified if there are no common factors, other than 1, in its numerator and denominator, a rational expression is simplified if it has no common factors, other than 1, in its numerator and denominator.

For example:

• $\frac{2}{3}$ is simplified because there are no common factors of 2 and 3.
• $\frac{2x}{3x}$ is not simplified because x is a common factor of 2 x and 3 x .

We use the Equivalent Fractions Property to simplify numerical fractions. We restate it here as we will also use it to simplify rational expression    s.

Notice that in the Equivalent Fractions Property, the values that would make the denominators zero are specifically disallowed. We see $b\ne 0,c\ne 0$ clearly stated. Every time we write a rational expression, we should make a similar statement disallowing values that would make a denominator zero. However, to let us focus on the work at hand, we will omit writing it in the examples.

Let’s start by reviewing how we simplify numerical fractions.

Simplify: $-\frac{36}{63}.$

Solution

Rewrite the numerator and denominator showing the common factors.
Simplify using the Equivalent Fractions Property.

Notice that the fraction $-\frac{4}{7}$ is simplified because there are no more common factors.

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Throughout this chapter, we will assume that all numerical values that would make the denominator be zero are excluded. We will not write the restrictions for each rational expression, but keep in mind that the denominator can never be zero. So in this next example, $x\ne 0$ and $y\ne 0$ .