Fractions with variables - simplifying and operating with algebraic fractions
A rational expression is a fraction where both numerator and denominator are polynomials.
Where P(x) and Q(x) are polynomials, and Q(x) ≠ 0.
Explore how changing the numerator and denominator affects the expression:
Domain: All real numbers except x = 1
Simplified: Already simplified
To simplify rational expressions:
Step 1: Factor numerator and denominator
\(x^2 - 4 = (x - 2)(x + 2)\)
\(x^2 - 2x - 8 = (x - 4)(x + 2)\)
Step 2: Write factored form
\(\frac{(x - 2)(x + 2)}{(x - 4)(x + 2)}\)
Step 3: Cancel common factors
\(\frac{x - 2}{\cancel{(x + 2)}} \cdot \frac{\cancel{(x + 2)}}{x - 4} = \frac{x - 2}{x - 4}\)
Simplified: \(\frac{x - 2}{x - 4}\)
Simplify the rational expression: \frac{x^2 - 9}{x^2 + 4x + 3}
1. \(\frac{x^2 - 1}{x^2 - x - 2} = \)
2. \(\frac{x^2 + 5x + 6}{x^2 + 2x - 3} = \)
3. \(\frac{2x + 4}{x^2 - 4} = \)
To add or subtract rational expressions, you need a common denominator.
\(\frac{a}{c} + \frac{b}{c} = \frac{a + b}{c}\)
\(\frac{a}{c} - \frac{b}{c} = \frac{a - b}{c}\)
\(\frac{a}{b} + \frac{c}{d} = \frac{a \cdot d + b \cdot c}{b \cdot d}\)
\(\frac{1}{x} + \frac{2}{x} = \)
\(\frac{x + 1}{x - 1} - \frac{x - 1}{x - 1} = \)
\(\frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d}\)
\(\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c} = \frac{a \cdot d}{b \cdot c}\)
To solve equations with rational expressions:
Step 1: Multiply both sides by (x - 1)
\(\frac{x + 1}{x - 1} \cdot (x - 1) = 2 \cdot (x - 1)\)
Step 2: Simplify
\(x + 1 = 2(x - 1)\)
Step 3: Solve
\(x + 1 = 2x - 2\)
\(x + 1 + 2 = 2x\)
\(x + 3 = 2x\)
\(3 = x\)
Check: Left side: \(\frac{3 + 1}{3 - 1} = \frac{4}{2} = 2\) ✓
Solve: \(\frac{2}{x} + \frac{3}{x + 1} = 1\)
The domain of a rational function excludes values that make the denominator zero.
Set denominator = 0: x² - 4 = 0
(x - 2)(x + 2) = 0
x = 2 or x = -2
Domain: All real numbers except x = -2 and x = 2
Find the domain of: \(\frac{x + 3}{x^2 - 9}\)