Strand 4 — Algebra
Algebraic Manipulation
5th Year · 6th Year (Leaving Cert)
- ✓By the end of this lesson students will be able to factorise algebraic expressions using various methods, including common factors, difference of two squares, and quadratic trinomials.
- ✓By the end of this lesson students will be able to simplify, add, subtract, multiply, and divide algebraic fractions.
- ✓By the end of this lesson students will be able to simplify surds and rationalise the denominator of expressions involving surds.
- ✓By the end of this lesson students will be able to apply algebraic manipulation techniques to solve problems in various mathematical contexts.
Key concepts
Factorisation is the process of writing an expression as a product of its factors. It is the reverse of expanding brackets. Key methods include: 1. Common Factor: Extracting the highest common factor from all terms. 2. Difference of Two Squares: Recognising expressions of the form a² - b². 3. Quadratic Trinomials: Factorising expressions of the form x² + bx + c or ax² + bx + c. 4. Grouping: For expressions with four terms, grouping terms to find common factors.
Algebraic fractions are fractions that contain algebraic expressions in the numerator, denominator, or both. Operations on algebraic fractions follow similar rules to numerical fractions: 1. Simplifying: Factorise the numerator and denominator, then cancel any common factors. 2. Multiplication: Multiply the numerators and multiply the denominators. 3. Division: Invert the second fraction and multiply. 4. Addition/Subtraction: Find a common denominator (usually the Lowest Common Multiple, LCM) and then add or subtract the numerators.
A surd is an irrational number that can be expressed as the root of an integer. Simplifying surds involves writing them in their simplest form, usually by extracting perfect square factors from under the square root sign. Operations with surds include: 1. Multiplication: √a * √b = √(ab) 2. Division: √a / √b = √(a/b) 3. Addition/Subtraction: Only 'like' surds (surds with the same number under the root sign) can be added or subtracted. 4. Rationalising the Denominator: Eliminating surds from the denominator of a fraction by multiplying both the numerator and denominator by an appropriate surd or its conjugate.
Key facts to remember
- 1Always look for a common factor first when factorising.
- 2The difference of two squares formula is a² - b² = (a - b)(a + b).
- 3To add or subtract algebraic fractions, find the Lowest Common Multiple (LCM) of the denominators.
- 4When multiplying algebraic fractions, multiply numerators and denominators directly.
- 5To divide algebraic fractions, invert the second fraction and then multiply.
- 6A surd is in its simplest form when the number under the root sign has no perfect square factors other than 1.
- 7Only 'like' surds can be added or subtracted.
- 8To rationalise a denominator of the form a/√b, multiply by √b/√b. For a/(√b ± √c), multiply by the conjugate (√b ∓ √c)/(√b ∓ √c).
Worked examples
Example 1
Factorise fully: 6x² + 11x - 10
Answer
(3x - 2)(2x + 5)
This method is often called 'factorising by grouping' or 'splitting the middle term'.
Example 2
Simplify the following algebraic expression: (x² - 4) / (x² + x - 6) + 2 / (x - 2)
Answer
(x² + 2x + 2) / [(x + 3)(x - 2)]
Always factorise fully before attempting to cancel or find a common denominator.
Example 3
Simplify √72 - √18 + (12 / √2) and express your answer in the form a√b.
Answer
9√2
Remember to rationalise the denominator if a surd is present in the denominator.
Common mistakes
- ✗Incorrectly cancelling terms in algebraic fractions (e.g., cancelling terms that are not factors). Only common factors can be cancelled.
- ✗Errors with signs when expanding brackets or combining terms, especially after finding a common denominator.
- ✗Not fully factorising expressions before simplifying algebraic fractions, leading to missed cancellations.
- ✗Attempting to add or subtract unlike surds (e.g., √2 + √3 ≠ √5).
- ✗Forgetting to multiply both the numerator and denominator when rationalising the denominator or finding a common denominator.
Exam tips
- ★Show all steps clearly, especially when dealing with multiple operations in algebraic fractions or surds. This helps in identifying errors and earning partial marks.
- ★Always check if your final answer can be simplified further, whether it's a factorised expression, an algebraic fraction, or a surd.
- ★Be meticulous with signs throughout your calculations, as a single sign error can lead to an incorrect final answer.
- ★Practise recognising the different factorisation patterns (common factor, difference of two squares, quadratic trinomials) quickly.
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