Welcome to Algebraic Manipulation!
Hello! Algebra might seem like a maze of letters and numbers, but think of it as learning the grammar of mathematics. Algebraic manipulation is the process of rearranging, simplifying, and transforming these mathematical sentences (expressions and equations) without changing their fundamental meaning.
Mastering this chapter is absolutely essential! These skills are the foundation for almost everything else you will do in IGCSE International Mathematics, from solving complex equations to graphing functions.
Section 1: Simplifying Expressions (Collecting Like Terms)
When we simplify an expression, we are tidying it up by combining terms that are mathematically similar.
What are Like Terms?
Like Terms are terms that contain the exact same variables raised to the exact same powers.
- Example: \(3x\) and \(7x\) are like terms.
- Example: \(5xy^2\) and \(-2xy^2\) are like terms.
- Not like terms: \(3x^2\) and \(3x\) (powers are different).
- Not like terms: \(4a\) and \(4b\) (variables are different).
Analogy: Sorting Fruit
Imagine your algebra expression is a mixed bag of fruit. You can only combine apples with apples, and bananas with bananas. You can't combine \(3\) apples and \(4\) bananas to get \(7\) 'apple-bananas'!
Example: Simplify \(2a + 3b + 5a - 9b\)
- Identify the 'A-terms': \(+2a\) and \(+5a\).
- Identify the 'B-terms': \(+3b\) and \(-9b\).
- Combine them: \((2a + 5a) + (3b - 9b)\)
- Result: \(7a - 6b\)
Quick Tip: Always include the sign (+ or -) in front of the term when moving it or combining it!
Key Takeaway for Simplification
Simplifying expressions involves collecting terms that have identical variable parts. If the variables or their powers don't match, you must keep them separate.
Section 2: Expanding Algebraic Expressions
Expanding means multiplying out the brackets to remove them. This gives you an equivalent expression in its simplest form.
1. Expanding Single Brackets (The Distributive Law)
Multiply the term outside the bracket by every term inside the bracket.
Example: Expand \(3x(2x - 4y)\)
Multiply \(3x\) by \(2x\): \(6x^2\)
Multiply \(3x\) by \(-4y\): \(-12xy\)
Result: \(6x^2 - 12xy\)
2. Expanding Double Brackets (FOIL/Grid Method)
To expand two brackets, you must ensure every term in the first bracket multiplies every term in the second bracket.
We use the famous mnemonic FOIL:
- First terms
- Outer terms
- Inner terms
- Last terms
Example: Expand \((2x + 1)(x - 4)\)
- First: \((2x) \times (x) = 2x^2\)
- Outer: \((2x) \times (-4) = -8x\)
- Inner: \((1) \times (x) = +x\)
- Last: \((1) \times (-4) = -4\)
Combine the results and collect the like terms (\(-8x + x\)):
Result: \(2x^2 - 7x - 4\)
3. Extended Content: Expanding Multiple Brackets
If you have three or more brackets, you must expand them in stages.
Example: Expand \((x - 2)(x + 3)(2x + 1)\)
- First, expand the first two brackets:
\((x - 2)(x + 3) = x^2 + 3x - 2x - 6 = x^2 + x - 6\) - Now, multiply this result by the third bracket:
\((x^2 + x - 6)(2x + 1)\) - Use the distributive method (or a 2x3 grid) to multiply every term:
\(x^2(2x) + x^2(1) + x(2x) + x(1) - 6(2x) - 6(1)\)
\(= 2x^3 + x^2 + 2x^2 + x - 12x - 6\) - Collect like terms: \(2x^3 + 3x^2 - 11x - 6\)
Key Takeaway for Expansion
Expansion is multiplication. Use FOIL for double brackets and remember to collect all like terms at the end to get the simplest answer.
Section 3: Factorising Algebraic Expressions
Factorising is the reverse of expanding: it involves putting brackets back into an expression. When asked to factorise, you must factorise fully (find the highest possible common factors).
1. Extracting Common Factors (Core & Extended)
This is the most common factorisation technique. Look for the largest number and the highest power of variables that divide into all terms.
Example: Factorise \(9x^2 + 15xy\)
- The numbers \(9\) and \(15\) share a common factor of \(3\).
- The variables share a common factor of \(x\).
- The Highest Common Factor (HCF) is \(3x\).
\(9x^2 + 15xy = 3x (3x + 5y)\)
2. Extended Content: Factorising Quadratics (\(ax^2 + bx + c\))
This involves reversing the FOIL process to find two binomials in brackets.
Example: Factorise \(x^2 + 7x + 12\)
We look for two numbers that:
- Multiply to give the last number (\(12\)).
- Add to give the middle number (\(7\)).
The numbers are \(3\) and \(4\). Result: \((x + 3)(x + 4)\)
3. Extended Content: Special Factorisation Forms
a) Difference of Two Squares (DOTS): \(a^2x^2 - b^2y^2\)
If you see two perfect squares separated by a minus sign, the pattern is:
$$\(A^2 - B^2 = (A - B)(A + B)\)$$
Example: Factorise \(a^2 - 4b^2\)
Here, \(A = a\) and \(B = 2b\).
Result: \((a - 2b)(a + 2b)\)
b) Factorisation by Grouping: \(ax + bx + kay + kby\)
This works for four-term expressions. Group the terms in pairs and find common factors in each pair.
Example: Factorise \(ax + bx + ay + by\)
- Group: \((ax + bx) + (ay + by)\)
- Factorise each pair: \(x(a + b) + y(a + b)\)
- Since \((a+b)\) is a common factor, factorise it out.
Result: \((x + y)(a + b)\)
Key Takeaway for Factorisation
Always check for the highest common factor first! Then, if it's a quadratic (three terms) or a DOTS (two terms), apply the appropriate structure.
Section 4: Indices II (Working with Powers)
Index laws provide rules for manipulating powers quickly. Remember, the base must be the same to use the multiplication and division rules!
The Basic Rules (Core & Extended)
- Multiplication Rule: When multiplying, add the powers.
$$\(a^m \times a^n = a^{m+n}\)$$ Example: \(6x^3 y^4 \times 5x^{-3} y^{-2} = 30x^{3+(-3)} y^{4+(-2)} = 30x^0 y^2 = 30y^2\) - Division Rule: When dividing, subtract the powers.
$$\(a^m \div a^n = a^{m-n}\)$$ Example: \(12a^5 \div 3a^{-2} = 4a^{5 - (-2)} = 4a^7\) - Power Rule: When raising a power to another power, multiply the powers.
$$\((a^m)^n = a^{mn}\)$$ Example: \((5x^3)^2 = 5^2 \times (x^3)^2 = 25x^6\)
Special Powers (Core & Extended)
- Zero Index: Anything (except 0) raised to the power of zero is 1.
$$\(a^0 = 1\)$$ - Negative Index: A negative index means taking the reciprocal (1 over the base, with a positive power).
$$\(a^{-n} = \frac{1}{a^n}\)$$ Example: Find the value of \(7^{-2}\). \(\frac{1}{7^2} = \frac{1}{49}\)
Extended Content: Fractional Indices (Roots)
A fractional index represents a root. The denominator of the fraction is the type of root, and the numerator is the power.
$$\(a^{\frac{1}{n}} = \sqrt[n]{a}\)$$ $$\(a^{\frac{m}{n}} = (\sqrt[n]{a})^m\)$$
Example: Calculate \(16^{\frac{1}{4}}\). This means the 4th root of 16, which is \(2\).
Example: Calculate \(8^{\frac{2}{3}}\). This means \((\sqrt[3]{8})^2 = (2)^2 = 4\)
Common Mistake to Avoid: When simplifying using indices, make sure you apply the rule to all parts of the term (numbers and variables).
Key Takeaway for Indices
Memorise the three basic rules (add, subtract, multiply powers). Remember that negative powers mean 'flip it' (reciprocal), and fractional powers mean 'root it'.
Section 5: Algebraic Fractions
Algebraic fractions follow the exact same rules as numerical fractions. The biggest challenge is dealing with the algebraic terms in the numerators and denominators.
1. Simplifying Algebraic Fractions (Core & Extended)
To simplify, look for common factors in the numerator (top) and the denominator (bottom) and cancel them out.
Core Example (One step required): Simplify \(\frac{x^2}{x^3}\)
\(\frac{x^2}{x^3} = \frac{x \times x}{x \times x \times x}\). Cancelling two \(x\)'s gives: \(\frac{1}{x}\)
Extended Example (Requires Factorisation): Simplify \(\frac{x^2 - 2x}{x^2 - 5x + 6}\)
- Factorise the numerator: \(x(x - 2)\)
- Factorise the denominator: \((x - 3)(x - 2)\)
- The expression is now: \(\frac{x(x - 2)}{(x - 3)(x - 2)}\)
- Cancel the common factor \((x - 2)\).
Result: \(\frac{x}{x - 3}\)
2. Extended Content: Multiplying and Dividing
Multiplying: Multiply tops together, multiply bottoms together. Simplify later.
Example: \(\frac{3a}{4} \times \frac{9a}{10} = \frac{27a^2}{40}\)
Dividing: Flip the second fraction and multiply (KCF: Keep, Change, Flip).
Example: \(\frac{3a}{4} \div \frac{9a}{10} = \frac{3a}{4} \times \frac{10}{9a}\)
Cancel common factors (\(3a\) in top and bottom, and 2 in 4 and 10): \(\frac{1}{2} \times \frac{5}{3} = \frac{5}{6}\)
3. Extended Content: Adding and Subtracting
You must find a Common Denominator (LCM) before adding or subtracting.
Example: Work out \(\frac{x}{3} + \frac{x - 4}{2}\)
- The LCM of 3 and 2 is 6.
- Convert both fractions:
\(\frac{x \times 2}{3 \times 2} + \frac{(x - 4) \times 3}{2 \times 3} = \frac{2x}{6} + \frac{3(x - 4)}{6}\) - Expand the numerator of the second fraction: \(\frac{2x + 3x - 12}{6}\)
- Simplify the numerator by collecting terms.
Result: \(\frac{5x - 12}{6}\)
Did you know? In the old days, algebra was sometimes called 'al-jabr', which is an Arabic term meaning 'the reunion of broken parts'—a perfect description for collecting terms and solving equations!
Key Takeaway for Algebraic Fractions
When multiplying/dividing, factorise and cancel first. When adding/subtracting, find the lowest common denominator first, then combine the numerators.