Mathematics (0580) | Introduction to algebra

👋 Welcome to the World of Algebra!

Welcome to the exciting world of Algebra! Don't worry if this chapter seems tricky at first—algebra is simply the mathematical shorthand we use to solve real-world problems more efficiently.

In this chapter, we will move from working with specific numbers to using letters (variables) to represent general numbers. This is the foundation of almost all higher mathematics! We will cover how to simplify, expand, and factorise expressions, and how to master the rules of powers (indices).

1. Variables, Expressions, and Substitution (C2.1/E2.1)

Algebra uses letters, called variables, to represent numbers that can change or are currently unknown.

What is the Difference?

• Expression: A mathematical phrase that contains numbers, variables, and operators (like +, -, x, /). It does not contain an equals sign.
  Example: \(3x + 5\)
• Formula: An equation that shows the relationship between two or more variables, often used in science or geometry.
  Example: The area of a triangle, \(A = \frac{1}{2}bh\).

1.1 Substitution: Swapping the Letter for a Number

Substitution is the process of replacing a variable in an expression or formula with a specific numerical value.

🔥 Top Tip: When you substitute a number for a letter, always put the number in brackets. This helps prevent mistakes with negative signs and powers.

Step-by-Step Example of Substitution

Find the value of the expression \(2a^2 - 3b\) if \(a = -4\) and \(b = 5\).

1. Substitute the values (using brackets):
   \(2(-4)^2 - 3(5)\)
2. Calculate the powers first (Remember BIDMAS/BODMAS):
   \((-4)^2 = (-4) \times (-4) = 16\)
   \(2(16) - 3(5)\)
3. Perform multiplication:
   \(32 - 15\)
4. Find the final answer:
   \(17\)

2. Algebraic Manipulation: Simplifying and Expanding (C2.2/E2.2)

2.1 Simplifying by Collecting Like Terms

To simplify an expression, we combine like terms. Like terms must have the exact same variable(s) raised to the exact same power.

Analogy: Think of algebra as shopping. You can only combine things that are exactly the same.
If you have 5 apples (\(5a\)), 2 bananas (\(2b\)), and you buy 3 more apples (\(+3a\)), you end up with 8 apples and 2 bananas.

Example: Simplify \(5x + 2y - x + 7y\)

• Group the \(x\) terms: \(5x - x = 4x\)
• Group the \(y\) terms: \(2y + 7y = 9y\)
• The simplified expression is: \(4x + 9y\)

⚠️ Common Mistake: Be careful with the sign! The sign always belongs to the term immediately following it.

Extended Example: Simplify \(2a^2 + 3ab - 1 + 5a^2 - 9ab + 4\)

• Collect \(a^2\) terms: \(2a^2 + 5a^2 = 7a^2\)
• Collect \(ab\) terms: \(3ab - 9ab = -6ab\)
• Collect number terms: \(-1 + 4 = 3\)
• Simplified expression: \(7a^2 - 6ab + 3\)

2.2 Expanding Products of Expressions

Expanding means removing brackets by multiplying every term inside the bracket by the term(s) outside the bracket (using the Distributive Law).

A. Single Brackets (Core & Extended)

Multiply the term outside by every term inside.

Example: Expand \(3x(2x - 4y)\)

• \(3x \times 2x = 6x^2\)
• \(3x \times (-4y) = -12xy\)
• Result: \(6x^2 - 12xy\)

B. Double Brackets (Core & Extended)

When multiplying two brackets, like \((A + B)(C + D)\), you must ensure every term in the first bracket multiplies every term in the second.

🧠 Memory Aid: FOIL (for expressions involving only one variable, as per Core syllabus note):

• First terms
• Outer terms
• Inner terms
• Last terms

Example: Expand \((2x + 1)(x - 4)\)

• F: \(2x \times x = 2x^2\)
• O: \(2x \times (-4) = -8x\)
• I: \(1 \times x = x\)
• L: \(1 \times (-4) = -4\)

Combine like terms (\(-8x + x\)): \(2x^2 - 7x - 4\)

C. Products of More than Two Brackets (Extended Only)

If you have three brackets, you must multiply the first two together first, then multiply the result by the third bracket.

Example: Expand \((x-2)(x+3)(2x+1)\).
(You would first multiply \((x-2)(x+3)\) to get \(x^2 + x - 6\). Then multiply this result by \((2x+1)\)).

3. Factorisation: The Reverse Process (C2.2/E2.2)

Factorising is the process of writing an expression as a product of its factors (putting the expression back into brackets). When asked to factorise, you must factorise fully.

3.1 Extracting Common Factors (Core & Extended)

Look for the Highest Common Factor (HCF) shared by all terms, both for numbers and variables.

Example 1 (Core): Factorise \(9x^2 + 15xy\)

• HCF of 9 and 15 is 3.
• HCF of \(x^2\) and \(xy\) is \(x\).
• Combined HCF is \(3x\).
• Result: \(3x(3x + 5y)\)

Example 2 (Extended - Simple Cubic): Factorise \(ax^3 + bx^2 + cx\)

• The common factor is \(x\).
• Result: \(x(ax^2 + bx + c)\)

3.2 Advanced Factorisation (Extended Only)

A. Factorisation by Grouping (E2.2)

Used for expressions with four terms, like \(ax + bx + kay + kby\). Group the terms in pairs and factorise each pair separately.

Example: Factorise \(3x + 6 + xy + 2y\)

1. Group terms: \((3x + 6) + (xy + 2y)\)
2. Factorise each group:
   \(3(x + 2) + y(x + 2)\)
3. Since \((x + 2)\) is common, factor it out:
   \((x + 2)(3 + y)\)

B. Difference of Two Squares (DOTS) (E2.2)

This applies to expressions where two square terms are separated by a minus sign: \(A^2 - B^2\).

Formula: \(A^2 - B^2 = (A - B)(A + B)\)

Example: Factorise \(a^2x^2 - b^2y^2\)

• This is \((ax)^2 - (by)^2\).
• Result: \((ax - by)(ax + by)\)

C. Factorising Quadratics (\(ax^2 + bx + c\)) (E2.2)

We look for two numbers that multiply to give \(c\) and add to give \(b\).

Example: Factorise \(x^2 + 7x + 12\)

• We need two numbers that multiply to 12 and add to 7. These are 3 and 4.
• Result: \((x + 3)(x + 4)\)

4. Working with Algebraic Fractions (C2.3/E2.3)

Algebraic fractions behave exactly like numerical fractions. The rules for simplification, addition, subtraction, multiplication, and division are the same.

4.1 Simplifying Algebraic Fractions (C2.3)

Core candidates are expected to simplify fractions requiring only one step (direct cancellation).

Example: Simplify \(\frac{x^2}{x}\)
\(\frac{x \times x}{x} = x\)

Example: Simplify \(\frac{3}{6x}\)
\(\frac{3}{3 \times 2x} = \frac{1}{2x}\)

4.2 Manipulating Algebraic Fractions (Extended Only) (E2.3)

A. Addition and Subtraction

You must find a common denominator before adding or subtracting.

Example: Work out \(\frac{x}{3} + \frac{x-4}{2}\)

1. Common denominator is 6.
2. Adjust the numerators:
   \(\frac{x \times 2}{3 \times 2} + \frac{(x-4) \times 3}{2 \times 3} = \frac{2x}{6} + \frac{3(x-4)}{6}\)
3. Combine and simplify:
   \(\frac{2x + 3x - 12}{6} = \frac{5x - 12}{6}\)

B. Multiplication and Division

• Multiplication: Multiply the numerators together and multiply the denominators together.
• Division: "Keep, Change, Flip." Keep the first fraction, change division to multiplication, and flip (take the reciprocal of) the second fraction.

C. Factorising and Simplifying Rational Expressions (E2.3)

For complex fractions, you must factorise the numerator and denominator first, and then cancel any common factors.

Example: Simplify \(\frac{x^2 - 2x}{x^2 - 5x + 6}\)

1. Factorise numerator: \(x(x - 2)\)
2. Factorise denominator (two numbers multiply to 6 and add to -5: -2 and -3): \((x - 2)(x - 3)\)
3. Rewrite the fraction:
   \(\frac{x(x - 2)}{(x - 2)(x - 3)}\)
4. Cancel the common factor \((x - 2)\).
5. Result: \(\frac{x}{x - 3}\)

5. Understanding Indices (Powers) (C2.4/E2.4)

An index (or power) tells you how many times a base number is multiplied by itself.

5.1 Definitions of Indices

Understanding these definitions is essential for both Core and Extended tiers.

1. Positive Integers: The standard power.
   Example: \(2^3 = 2 \times 2 \times 2 = 8\)
2. Zero Index: Any non-zero number raised to the power of zero is 1.
   Rule: \(a^0 = 1\)
   Example: \((12xy)^0 = 1\)
3. Negative Integers: A negative index means taking the reciprocal (1 divided by the base raised to the positive power).
   Rule: \(a^{-n} = \frac{1}{a^n}\)
   Example: \(7^{-2} = \frac{1}{7^2} = \frac{1}{49}\)
4. Fractional Indices (Extended Only): The denominator of the fraction represents the root, and the numerator represents the power.
   Rule: \(a^{m/n} = (\sqrt[n]{a})^m\)
   Example: \(8^{1/3} = \sqrt[3]{8} = 2\).

5.2 The Rules of Indices (C2.4/E2.4)

These rules help you simplify expressions quickly.

Rule 1: Multiplication (Adding Powers)

When multiplying terms with the same base, add the powers.

Rule: \(a^m \times a^n = a^{m+n}\)
Example: \(x^4 \times x^3 = x^{4+3} = x^7\)

Rule 2: Division (Subtracting Powers)

When dividing terms with the same base, subtract the powers.

Rule: \(a^m \div a^n = a^{m-n}\)
Example (Core): \(2^3 \div 2^4 = 2^{3-4} = 2^{-1} = \frac{1}{2}\)
Example (Extended): \(12a^5 \div 3a^{-2} = 4a^{5 - (-2)} = 4a^7\)

Rule 3: Power of a Power (Multiplying Powers)

When raising a power to another power, multiply the indices.

Rule: \((a^m)^n = a^{mn}\)
Example: \((5x^3)^2 = 5^2 \times (x^3)^2 = 25x^6\)

Rule 4: Power of a Product

Raise every factor inside the bracket to the power.

Rule: \((ab)^n = a^n b^n\)
Example: \((6x^2y)^3 = 6^3 x^{2 \times 3} y^3 = 216x^6y^3\)

5.3 Solving Simple Index Equations (Extended Only) (E2.4)

To solve equations where the unknown is the power, try to rewrite both sides of the equation using the same base number.

Example 1: Solve \(2^x = 32\)

Rewrite 32 as a power of 2: \(32 = 2^5\)
\(2^x = 2^5\)
Since the bases are the same, the powers must be equal: \(x = 5\)

Example 2: Solve \(5^{x+1} = 25^x\)

Rewrite 25 as a power of 5: \(25 = 5^2\)
\(5^{x+1} = (5^2)^x\)
Using Rule 3 on the right side: \(5^{x+1} = 5^{2x}\)
Equate the powers: \(x + 1 = 2x\)
Solve for \(x\): \(1 = x\)

🏁 Final Review: Core Algebra Checklist

If you are targeting Core grades, make sure you are confident with:

• Substituting numbers into expressions and simple formulas (C2.1).
• Simplifying expressions by collecting like terms, including positive and negative coefficients (C2.2.1).
• Expanding single brackets and products of two brackets involving one variable (C2.2.2).
• Factorising by extracting common factors fully (C2.2.3).
• Simplifying algebraic fractions where only one cancellation step is required (C2.3).
• Using positive, zero, and negative integer indices and the basic rules of indices (C1.7 & C2.4).