CIE IGCSE Mathematics (0580) Algebraic fractions Study Notes - New Syllabus
CIE IGCSE Mathematics (0580) Algebraic fractions Study Notes
LEARNING OBJECTIVE
- Manipulating Algebraic Fractions
Key Concepts:
- Manipulating Algebraic Fractions
Manipulating Algebraic Fractions
Manipulating Algebraic Fractions
Algebraic fractions are expressions that contain variables in the numerator, the denominator, or both. The same principles used in numeric fraction operations apply — such as finding common denominators, multiplying/dividing across, and simplifying using algebraic identities and factorization.
Key Concepts:
- When adding or subtracting algebraic fractions, find the least common denominator (LCD).
- To multiply, multiply the numerators together and the denominators together.
- To divide, multiply the first fraction by the reciprocal of the second.
- Always simplify your final answer by cancelling any common factors.
Important Tips:
- Use factorization techniques to simplify complex expressions.
- Be cautious with signs while expanding and simplifying.
- Ensure denominators are never zero (undefined expressions).
Example:
Simplify: \( \frac{x}{3} + \frac{x – 4}{2} \)
▶️ Answer/Explanation
LCM of 3 and 2 is 6.
\( \frac{x}{3} = \frac{2x}{6}, \quad \frac{x – 4}{2} = \frac{3(x – 4)}{6} \)
\( = \frac{2x + 3(x – 4)}{6} = \frac{2x + 3x – 12}{6} = \frac{5x – 12}{6} \)
Final Answer: \( \frac{5x – 12}{6} \)
Example:
Simplify: \( \frac{2x}{3} – \frac{3(x – 5)}{2} \)
▶️ Answer/Explanation
LCM of 3 and 2 is 6.
\( \frac{2x}{3} = \frac{4x}{6}, \quad \frac{3(x – 5)}{2} = \frac{9(x – 5)}{6} \) \( = \frac{4x – 9(x – 5)}{6} = \frac{4x – 9x + 45}{6} = \frac{-5x + 45}{6} \)
Final Answer: \( \frac{-5x + 45}{6} \)
Example:
Simplify: \( \frac{3a}{4} \times \frac{9a}{10} \)
▶️ Answer/Explanation
\( \frac{3a}{4} \times \frac{9a}{10} = \frac{3 \times 9 \times a \times a}{4 \times 10} = \frac{27a^2}{40} \)
Final Answer: \( \frac{27a^2}{40} \)
Example:
Simplify: \( \frac{3a}{4} \div \frac{9a}{10} \)
▶️ Answer/Explanation
To divide, multiply by the reciprocal:
\( \frac{3a}{4} \div \frac{9a}{10} = \frac{3a}{4} \times \frac{10}{9a} \)
Cancel \( a \):
\( = \frac{3}{4} \times \frac{10}{9} = \frac{30}{36} = \frac{5}{6} \)
Final Answer: \( \frac{5}{6} \)
Example:
Simplify: \( \frac{1}{x – 2} + \frac{x + 1}{x – 3} \)
▶️ Answer/Explanation
Use a common denominator:
\( \text{LCD} = (x – 2)(x – 3) \)
\( \frac{1}{x – 2} = \frac{x – 3}{(x – 2)(x – 3)}, \quad \frac{x + 1}{x – 3} = \frac{(x + 1)(x – 2)}{(x – 3)(x – 2)} \)
Add: \( \frac{x – 3 + (x + 1)(x – 2)}{(x – 2)(x – 3)} \)
Expand numerator: \( (x + 1)(x – 2) = x^2 – x – 2 \)
Then: \( \text{Numerator} = x – 3 + x^2 – x – 2 = x^2 – 5 \)
Final: \( \frac{x^2 – 5}{(x – 2)(x – 3)} \)
Factorising and Simplifying Rational Expressions
Factorising and Simplifying Rational Expressions
A rational expression is a fraction that contains polynomials in the numerator, denominator, or both. To simplify rational expressions:
- Factor both the numerator and the denominator completely.
- Cancel any common factors that appear in both the numerator and denominator.
- Check for restrictions: the values that make the denominator zero must be excluded.
Example:
Simplify \( \dfrac{x^2 – 2x}{x^2 – 5x + 6} \)
▶️ Answer/Explanation
Step 1: Factor numerator and denominator.
Numerator: \( x^2 – 2x = x(x – 2) \)
Denominator: \( x^2 – 5x + 6 = (x – 2)(x – 3) \)
Step 2: Cancel common factors.
\( \dfrac{x(x – 2)}{(x – 2)(x – 3)} = \dfrac{x}{x – 3} \)
Answer: \( \dfrac{x}{x – 3} \), with \( x \ne 2, 3 \)
Example:
Simplify \( \dfrac{x^2 – 9}{x^2 + 5x + 6} \)
▶️ Answer/Explanation
Step 1: Factor both expressions.
Numerator: \( x^2 – 9 = (x – 3)(x + 3) \)
Denominator: \( x^2 + 5x + 6 = (x + 2)(x + 3) \)
Step 2: Cancel common factors.
\( \dfrac{(x – 3)(x + 3)}{(x + 2)(x + 3)} = \dfrac{x – 3}{x + 2} \)
Answer: \( \dfrac{x – 3}{x + 2} \), with \( x \ne -2, -3 \)