Let's dive into the world of rational expressions and rational numbers. While both involve fractions, they operate in different mathematical landscapes. Understanding their definitions and differences is key to mastering algebra and number theory.
A rational number is any number that can be expressed as the quotient or fraction $\frac{p}{q}$ of two integers, a numerator $p$ and a non-zero denominator $q$. Since $q$ can be equal to $1$, every integer is a rational number.
A rational expression is an algebraic expression that can be written as the ratio of two polynomials, $\frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are polynomials and $Q(x)$ is not the zero polynomial.
| Feature | Rational Numbers | Rational Expressions |
|---|---|---|
| Definition | Ratio of two integers. | Ratio of two polynomials. |
| Variables | Do not contain variables. | Contain variables (usually $x$). |
| Form | $\frac{p}{q}$, where $p$ and $q$ are integers. | $\frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are polynomials. |
| Examples | $\frac{2}{3}$, $-5$, $0.25$ | $\frac{x}{x+1}$, $\frac{x^2 - 1}{x+2}$ |
| Simplification | Simplification involves reducing the fraction to its lowest terms by dividing out common factors. | Simplification involves factoring polynomials and canceling common factors. |
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