People's Education Press: Junior High School Mathematics, Grade 8, Part 1: Introduction to Rational Expressions: Defining Concepts, Exploring Meaning, and Core Properties

Imagine you have two complex-shaped plots of land. You need a unified formula to describe their area ratio. When such a ratio can no longer be expressed using simple integers (like $\frac{3}{4}$), but requires introducing variables (such as $x$) to capture the underlying patterns, we move from fractions into the fascinating world of rational expressions.fractionsintorational expressionsthe magical realm. Rational expressions are mathematics' 'advanced language,' granting letters in the denominator the freedom to 'dance,' allowing us to model more complex quantitative relationships in the real world.

1. Defining Rational Expressions: The Home for Variables

A rational expression is not merely a pile of two polynomials. Its core essence lies inthe denominator. If we write a rational expression in the form $\frac{A}{B}$, then $A$ and $B$ must be polynomials, and the key point is:the denominator $B$ must contain a variable. This is the sole criterion for distinguishing between polynomials and rational expressions.

2. Exploring Meaning: The Forbidden 'Zero Zone'

In the realm of mathematics, a zero denominator is an absolute taboo. Therefore, for the rational expression $\frac{A}{B}$ to be meaningfulrequires the prerequisite that $B \neq 0$. This condition acts like a safety barrier, ensuring the rigor of algebraic logic. When discussing when a rational expression equals zero, we must satisfy the dual standard: the numerator is zero and the denominator is not zero.

Identification Technique

To determine if an expression is a rational expression, first check if it has the outer shell of $\frac{A}{B}$, then scan the denominator. If the denominator contains only constants or $\pi$, it remains a polynomial; if the denominator contains variables like $x$, $a$, or $t$, it is a rational expression.

3. Core Properties: The Magic of Equivalence

The core property of rational expressions is an evolution of fraction properties: the value of a rational expression remains unchanged when its numerator and denominator are multiplied or divided by the samenon-zeropolynomial. This is the logical foundation forsimplifying (reducing)(simplifying)finding a common denominator(performing operations with a common denominator).

🎯 Core Rules

1. Form: $\frac{A}{B}$ (where $A$ and $B$ are polynomials, and $B$ contains a variable);
2. Constraint: $B \neq 0$ for the expression to be meaningful;
3. Essence: The value remains unchanged when both numerator and denominator change together.

$\frac{A}{B} = \frac{A \cdot M}{B \cdot M} \quad (M \neq 0)$

QUESTION 1

[Fill in the blanks] 1. Fill in the blanks and determine whether the resulting expression is a rational expression: (1) A writer spends $m$ days writing the first part of a novel, and $n$ days writing the second part. The entire novel is 1.2 million characters long. The average daily writing output is ______; (2) Walking a 10 km route takes $2x$ hours. Cycling time is 0.2 hours less than half the walking time. The cycling speed is ______; (3) Person A needs $t$ hours to complete a task. Person B takes 1 hour less than A. The total work is 1 unit. B's efficiency is ______.

(1) $\frac{120}{m+n}$ (rational expression); (2) $\frac{10}{x-0.2}$ (rational expression); (3) $\frac{1}{t-1}$ (rational expression)

(1) $120(m+n)$ (polynomial); (2) $\frac{10}{x}$ (rational expression); (3) $\frac{1}{t}$ (rational expression)

QUESTION 2

[Short Answer] 3. For what values of $x$ are the following rational expressions defined? (1) $\frac{1}{3x}$; (2) $\frac{1}{3-x}$; (3) $\frac{x-5}{3x+5}$; (4) $\frac{1}{x^2-16}$.

(1) $x \neq 0$; (2) $x \neq 3$; (3) $x \neq -\frac{5}{3}$; (4) $x \neq \pm 4$

(1) $x=0$; (2) $x=3$; (3) $x=-\frac{5}{3}$; (4) $x=4$

QUESTION 3

[Short Answer] Simplify: (1) $\frac{2bc}{ac}$; (2) $\frac{(x+y)y}{xy^2}$; (3) $\frac{x^2+xy}{(x+y)^2}$; (4) $\frac{x^2-y^2}{(x-y)^2}$.

(1) $\frac{2b}{a}$; (2) $\frac{x+y}{xy}$; (3) $\frac{x}{x+y}$; (4) $\frac{x+y}{x-y}$

(1) $\frac{2b}{c}$; (2) $\frac{y}{x}$; (3) $\frac{1}{x+y}$; (4) $\frac{x-y}{x+y}$

QUESTION 4

[Short Answer] 1. Write the results of Problem 1 and Problem 2 in Section 15.2.1.

Problem 1: $\frac{s}{a}$; Problem 2: $\frac{v}{a}$

Problem 1: $sa$; Problem 2: $v-a$

QUESTION 5

[Fill in the blanks] 1. Fill in the blanks: (1) $3^0 = \_\_\_$, $3^{-2} = \_\_\_$; (2) $(-3)^0 = \_\_\_$, $(-3)^{-2} = \_\_\_$; (3) $b^0 = \_\_\_$, $b^{-2} = \_\_\_ (b \neq 0)$.

(1) $1, \frac{1}{9}$; (2) $1, \frac{1}{9}$; (3) $1, \frac{1}{b^2}$

(1) $0, -9$; (2) $0, -9$; (3) $0, -b^2$