Elementary Algebra

Lynn Marecek

ISBN #9780998625713

1st Edition

6,645 Questions

32,192 Students Helped

Homework Questions

Summary

Learning Objectives

Key Concepts

Example Problems

Explanations

Common Mistakes

Summary

This chapter emphasizes understanding and working with rational expressions. Key skills include determining when an expression is undefined, evaluating expressions by substituting values, and simplifying expressions by factoring and canceling common factors. Special attention is given to handling opposites, which often simplify directly to -1, and to performing multiplication and division with rational expressions similar to numerical fractions.

Learning Objectives

Determine the values for which a given rational expression is undefined by solving for when the denominator equals zero.

Evaluate rational expressions by substituting given values and simplifying the result.

Simplify rational expressions by factoring polynomials and canceling common factors, including handling opposite factors.

Multiply and divide rational expressions by applying the properties of fraction multiplication and division.

Key Concepts

CONCEPT

DEFINITION

Rational Expression

An expression of the form p(x)/q(x), where p(x) and q(x) are polynomials, and q(x) is not equal to zero.

Undefined Expression

A rational expression is undefined for any variable values that make the denominator equal to zero.

Equivalent Fractions Property

The property stating that if a, b, and c are numbers with b ≠ 0 and c ≠ 0 then a/b = (a * c)/(b * c); used to simplify fractions.

Opposite Factors

Factors such as (a - b) and (b - a) which are negatives of each other, so their quotient simplifies to -1.

Factoring

The process of rewriting a polynomial as a product of its factors, which is essential for simplifying rational expressions.

Example Problems

Example 1

In the following exercises, determine the values for which the rational expression is undefined. (a) $\frac{2 x}{z}$ (b) $\frac{4 p-1}{6 p-5}$ (c) $\frac{n-3}{n^{2}+2 n-8}$

Example 2

In the following exercises, determine the values for which the rational expression is undefined. (a) $\frac{10 m}{11 n}$ (b) $\frac{6 y+13}{4 y-9}$ (c) $\frac{b-8}{b^{2}-36}$

Example 3

In the following exercises, determine the values for which the rational expression is undefined. (a) $\frac{4 x^{2} y}{3 y}$ (b) $\frac{3 x-2}{2 x+1}$ (c) $\frac{u-1}{u^{2}-3 u-28}$

Example 4

In the following exercises, determine the values for which the rational expression is undefined. (a) $\frac{5 p q^{2}}{9 q}$ (b) $\frac{7 a-4}{3 a+5}$ (c) $\frac{1}{x^{2}-4}$

Example 5

In the following exercises, evaluate the rational expression for the given values. $\frac{2 x}{x-1}$ (a) $x=0$ (b) $x=2$ (c) $x=-1$

Step-by-Step Explanations

QUESTION

For the rational expression (x + 4)/(x^2 + 5x + 6), for which values of x is the expression undefined?

STEP-BY-STEP ANSWER:

Step 1: Factor the denominator: x^2 + 5x + 6 factors as (x + 2)(x + 3).
Step 2: Set each factor equal to zero: x + 2 = 0 and x + 3 = 0.
Step 3: Solve the equations: x = -2 and x = -3.
Final Answer: The expression is undefined for x = -2 and x = -3.

Determine Undefined Values

QUESTION

Evaluate the expression (2x + 3)/(3x - 5) for x = 2.

STEP-BY-STEP ANSWER:

Step 1: Substitute x = 2 into the expression: (2(2) + 3)/(3(2) - 5).
Step 2: Calculate the numerator: 4 + 3 = 7.
Step 3: Calculate the denominator: 6 - 5 = 1.
Final Answer: The evaluated result is 7/1 = 7.

Evaluate a Rational Expression

QUESTION

Simplify the expression (x - 8)/(8 - x).

STEP-BY-STEP ANSWER:

Step 1: Recognize that (8 - x) is the opposite of (x - 8), i.e., 8 - x = -(x - 8).
Step 2: Rewrite the expression as (x - 8)/[-(x - 8)].
Step 3: Cancel the common factor (x - 8), keeping in mind the negative sign.
Final Answer: The simplified expression equals -1.

Simplify Rational Expressions with Opposite Factors

Common Mistakes

Failing to determine and state the variable values that make the denominator zero, leading to incorrect solution domains.
Incorrectly canceling terms across addition or subtraction (canceling parts of a sum rather than a product).
Mismanaging negative signs, especially when dealing with opposite factors like (a - b) and (b - a).
Not fully factoring polynomials before canceling common factors, which can lead to incomplete simplification.