Question
Formula for rational expression and equation
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Answered by
GPT-4o mini
A **rational expression** is a fraction in which both the numerator and the denominator are polynomials. The general form of a rational expression can be written as:
\[
R(x) = \frac{P(x)}{Q(x)}
\]
where \( P(x) \) and \( Q(x) \) are polynomials, and \( Q(x) \neq 0 \) (since division by zero is undefined).
### Examples of Rational Expressions
1. \(\frac{x^2 - 1}{x + 2}\)
2. \(\frac{3x + 4}{x^2 - 5x + 6}\)
3. \(\frac{2}{x^3 - 4}\)
### Properties of Rational Expressions
1. **Domain**: The domain of a rational expression is all real numbers except where the denominator is equal to zero.
2. **Simplification**: Rational expressions can often be simplified by factoring the numerator and the denominator and canceling common factors.
3. **Addition and Subtraction**: To add or subtract rational expressions, you need a common denominator.
### Rational Equations
A **rational equation** is an equation that involves rational expressions. An example of a rational equation is:
\[
\frac{x + 1}{x - 2} = \frac{2}{x + 3}
\]
### Solving Rational Equations
To solve a rational equation, follow these steps:
1. **Identify the Domain**: Determine values that make the denominators zero, as these will not be in the domain.
2. **Find a Common Denominator**: Multiply both sides of the equation by the least common denominator (LCD) to eliminate the fractions.
3. **Solve the Resulting Equation**: This will often lead to a polynomial equation which can be solved using standard algebraic techniques.
4. **Check for Extraneous Solutions**: Any solutions found should be checked against the original equation to ensure they do not make any denominators zero.
### Example of a Rational Equation Solution
Consider the equation:
\[
\frac{2x}{x + 1} = \frac{3}{x - 2}
\]
**Step 1: Identify the Domain**
The denominators \( x + 1 \) and \( x - 2 \) must not equal zero, so:
- \( x \neq -1 \)
- \( x \neq 2 \)
**Step 2: Find the Common Denominator**
The common denominator would be \( (x + 1)(x - 2) \).
**Step 3: Multiply Through by the LCD:**
\[
2x(x - 2) = 3(x + 1)
\]
**Step 4: Solve the Resulting Equation**
Expanding both sides:
\[
2x^2 - 4x = 3x + 3
\]
Rearranging gives:
\[
2x^2 - 7x - 3 = 0
\]
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[
x = \frac{7 \pm \sqrt{(-7)^2 - 4 \cdot 2 \cdot (-3)}}{2 \cdot 2}
\]
\[
x = \frac{7 \pm \sqrt{49 + 24}}{4} = \frac{7 \pm \sqrt{73}}{4}
\]
**Step 5: Check Solutions**
Check whether \( \frac{7 + \sqrt{73}}{4} \) or \( \frac{7 - \sqrt{73}}{4} \) equal \(-1\) or \(2\) — both must be checked with the original expression to ensure they are part of the domain. If they are not, discard those solutions.
Using these concepts, you can effectively work with rational expressions and equations!
\[
R(x) = \frac{P(x)}{Q(x)}
\]
where \( P(x) \) and \( Q(x) \) are polynomials, and \( Q(x) \neq 0 \) (since division by zero is undefined).
### Examples of Rational Expressions
1. \(\frac{x^2 - 1}{x + 2}\)
2. \(\frac{3x + 4}{x^2 - 5x + 6}\)
3. \(\frac{2}{x^3 - 4}\)
### Properties of Rational Expressions
1. **Domain**: The domain of a rational expression is all real numbers except where the denominator is equal to zero.
2. **Simplification**: Rational expressions can often be simplified by factoring the numerator and the denominator and canceling common factors.
3. **Addition and Subtraction**: To add or subtract rational expressions, you need a common denominator.
### Rational Equations
A **rational equation** is an equation that involves rational expressions. An example of a rational equation is:
\[
\frac{x + 1}{x - 2} = \frac{2}{x + 3}
\]
### Solving Rational Equations
To solve a rational equation, follow these steps:
1. **Identify the Domain**: Determine values that make the denominators zero, as these will not be in the domain.
2. **Find a Common Denominator**: Multiply both sides of the equation by the least common denominator (LCD) to eliminate the fractions.
3. **Solve the Resulting Equation**: This will often lead to a polynomial equation which can be solved using standard algebraic techniques.
4. **Check for Extraneous Solutions**: Any solutions found should be checked against the original equation to ensure they do not make any denominators zero.
### Example of a Rational Equation Solution
Consider the equation:
\[
\frac{2x}{x + 1} = \frac{3}{x - 2}
\]
**Step 1: Identify the Domain**
The denominators \( x + 1 \) and \( x - 2 \) must not equal zero, so:
- \( x \neq -1 \)
- \( x \neq 2 \)
**Step 2: Find the Common Denominator**
The common denominator would be \( (x + 1)(x - 2) \).
**Step 3: Multiply Through by the LCD:**
\[
2x(x - 2) = 3(x + 1)
\]
**Step 4: Solve the Resulting Equation**
Expanding both sides:
\[
2x^2 - 4x = 3x + 3
\]
Rearranging gives:
\[
2x^2 - 7x - 3 = 0
\]
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[
x = \frac{7 \pm \sqrt{(-7)^2 - 4 \cdot 2 \cdot (-3)}}{2 \cdot 2}
\]
\[
x = \frac{7 \pm \sqrt{49 + 24}}{4} = \frac{7 \pm \sqrt{73}}{4}
\]
**Step 5: Check Solutions**
Check whether \( \frac{7 + \sqrt{73}}{4} \) or \( \frac{7 - \sqrt{73}}{4} \) equal \(-1\) or \(2\) — both must be checked with the original expression to ensure they are part of the domain. If they are not, discard those solutions.
Using these concepts, you can effectively work with rational expressions and equations!