Definition of Rational Expression

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Algebra II › Definition of Rational Expression

Questions 1 - 4

1

Simplify:

$\frac{x^2-1}{x^2+4x+2}\div\frac{x^2+2x+1}{x^2+3x+2}$

$\frac{x-1}{x+2}$

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

$\frac{1}{\frac{1}{x+1}}$

$\frac{x+2}{x-1}$

Explanation

This problem is a lot simpler if we factor all the expressions involved before proceeding:

$\frac{x^2-1}{x^2+4x+2}\div\frac{x^2+2x+1}{x^2+3x+2} \vspace{5mm}\ = \frac{(x+1)(x-1)}{(x+2)(x+2)}\div\frac{(x+1)(x+1)}{(x+2)(x+1)}$

Next let's remember how we divide one fraction by another—by multiplying by the reciprocal:

$\frac{(x+1)(x-1)}{(x+2)(x+2)}\div\frac{(x+1)(x+1)}{(x+2)(x+1)} \vspace{5mm}\ = \frac{(x+1)(x-1)}{(x+2)(x+2)}\times\frac{(x+2)(x+1)}{(x+1)(x+1)}$

In this form, we can see that a lot of terms are going to start canceling with each other. All that we're left with is just $\frac{x-1}{x+2}$ .

2

Which of the following is the best definition of a rational expression?

$\textup{The ratio of two polynomials that cannot have a zero denominator.}$

$\textup{The ratio of two polynomials that must be different from each other.}$

$\textup{The ratio of two fractions that may have a zero denominator.}$

$\textup{The ratio of any two real numbers.}$

$\textup{The ratio of any two imaginary numbers.}$

Explanation

The rational expression is a ratio of two polynomials.

$\frac{A(x)}{B(x)}$

The denominator cannot be zero.

An example of a rational expression is:

$\frac{x^2+x-5}{x+8}$

The answer is:

$\textup{The ratio of two polynomials that cannot have a zero denominator.}$

3

Determine the domain of

$\frac{x+1}{x-1}$

All real numbers

Explanation

Because the denominator cannot be zero, the domain is all other numbers except for 1, or

4

Which of the following fractions is NOT equivalent to $- \frac{x-5}{2x + 3}$ ?

$\frac{x+5}{2x + 3}$

$\frac{-x+5}{2x+3}$

$\frac{x-5}{-2x-3}$

$\frac{\left -(x-5 \right )}{2x+3}$

$\frac{x-5}{\left -(2x+3 \right )}$

Explanation

We know that $-\frac{a}{b}$ is equivalent to $\frac{-a}{b}$ or $\frac{a}{-b}$ .

By this property, there is no way to get $\frac{x+5}{2x+3}$ from $-\frac{x-5}{2x+3}$ .

Therefore the correct answer is $\frac{x+5}{2x+3}$ .

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