ERF - Rational Expressions and Operations Lesson

Rational Expressions and Operations

Rational Expressions

A rational expression is a ratio of two polynomial expressions.

$LaTeX: \frac{P(x)}{Q(x)}$ $\frac{P(x)}{Q(x)}$ where P(x) and Q(x) are polynomials and $LaTeX: Q(x)\ne0$ $Q(x)\ne0$ because that would make the fraction undefined.

It's simply a fraction with polynomials! Here are some examples: $LaTeX: \frac{6}{x-1},\frac{z^2-1}{z^2+5},\frac{4x^2+6x-10}{1},\frac{m^4+18m+1}{m^2-m-6}$ $\frac{6}{x-1},\frac{z^2-1}{z^2+5},\frac{4x^2+6x-10}{1},\frac{m^4+18m+1}{m^2-m-6}$

Similar to rational numbers, rational expressions can be proper or improper.

Excluded Values

We must be careful to avoid division by zero because it creates an undefined expression. (Remember: the denominator can never equal zero.) These values that create an undefined expression are called the excluded values. We set the denominator equal to zero to find the excluded values. When we look at the graph of the function, we see that the excluded values are where the domain is restricted by the asymptotes or holes. We'll talk more about that later but now, let's take a look at these examples of how to find excluded values.

Rational Expression:	Set denominator equal to zero and solve for x to find the excluded value.	Graph of the rational function:
$LaTeX: \frac{1}{x}$ $\frac{1}{x}$	$LaTeX: x\ne0$ $x\ne0$ Therefore, x = 0 is the excluded value. Domain: $LaTeX: \left\lbrace x\in\mathbb{R}\vert x\ne0\right\rbrace$ $\left\lbrace x\in\mathbb{R}\vert x\ne0\right\rbrace$ (All real numbers, except 0) Domain written in interval notation: $LaTeX: (-\infty,0)\cup(0,\infty)$ $(-\infty,0)\cup(0,\infty)$	$LaTeX: f(x)=\frac{1}{x}$ $f(x)=\frac{1}{x}$
$LaTeX: \frac{x+3}{x-4}$ $\frac{x+3}{x-4}$	x - 4 = 0 x = 4 Therefore, x = 4 is our excluded value. Domain: $LaTeX: \left\lbrace x\in\mathbb{R}\vert x\ne4\right\rbrace$ $\left\lbrace x\in\mathbb{R}\vert x\ne4\right\rbrace$ (All real numbers, except 4) Domain written in interval notation: $LaTeX: (-\infty,4)\cup(4,\infty)$ $(-\infty,4)\cup(4,\infty)$	$LaTeX: f(x)=\frac{x+3}{x-4}$ $f(x)=\frac{x+3}{x-4}$
$LaTeX: \frac{x+6}{x^2+2x-3}$ $\frac{x+6}{x^2+2x-3}$	$x^2+2x-3$ Factor the expression and set it equal to zero. (x+3)(x-1) = 0 Solve for x. x = -3 and x = 1 There are two excluded values in this example: x = -3 and x ≠ 1 Domain: $LaTeX: \left\lbrace x\in\mathbb{R}\vert x\ne-3,x\ne1\right\rbrace$ $\left\lbrace x\in\mathbb{R}\vert x\ne-3,x\ne1\right\rbrace$ (All real numbers, except -3 and 1) Domain written in interval notation: $LaTeX: (-\infty,-3)\cup(-3,1)\cup(1,\infty)$ $(-\infty,-3)\cup(-3,1)\cup(1,\infty)$	$LaTeX: f(x)=\frac{x+6}{x^2+2x-3}$ $f(x)=\frac{x+6}{x^2+2x-3}$

Below are three videos with more examples of finding excluded values. Make sure you take notes as you watch each one.

Simplifying Rational Expressions

Simplifying a rational expression means to reduce the fraction to lowest terms. These are the steps to follow when simplifying rational expressions.

Factor the numerator and denominator completely.
List excluded values.
Cancel common factors.

Example

equation image indicator $LaTeX: \frac{y^2-25}{y^2-3y-40}=\frac{\left(y+5\right)\left(y-5\right)}{\left(y-8\right)\left(y+5\right)}=\frac{\left(y-5\right)}{\left(y-8\right)}=y\ne-5,\:8$ $\frac{y^2-25}{y^2-3y-40}=\frac{\left(y+5\right)\left(y-5\right)}{\left(y-8\right)\left(y+5\right)}=\frac{\left(y-5\right)}{\left(y-8\right)}=y\ne-5,\:8$

The information just discussed will help you greatly in simplifying rational expressions.

Now, to simplify a rational expression:

Factor the numerator and denominator completely.
Simplify the monomials (use exponent rules).
Eliminate common binomial factors.

The following 8 videos will discuss and model in greater detail the concepts surrounding the simplification of rational expressions. Take notes as you watch them; then complete the practice activity to master these concepts.

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