Algebra

Algebraic Expressions – Revision Notes

Edexcel IGCSE Mathematics A

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Key Definitions and Terminology

• **Variable**: A letter (such as *x*, *y*, or *n*) used to represent an unknown or changeable quantity.
• **Term**: A single number, variable, or the product of numbers and variables. For example, in the expression `3x² + 5x − 7`, the terms are `3x²`, `5x`, and `−7`.
• **Coefficient**: The numerical part of a term that multiplies the variable. In `3x²`, the coefficient is **3**.
• **Constant**: A term with no variable attached. In `3x² + 5x − 7`, the constant is **−7**.
• **Expression**: A collection of terms connected by addition or subtraction. Unlike an equation, an expression has **no equals sign**.
• **Like Terms**: Terms that have exactly the same variable(s) raised to the same power(s). For example, `4x` and `−2x` are like terms; `4x` and `4x²` are **not**.

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Main Concepts

1. Simplifying Expressions by Collecting Like Terms

Combine terms that share the same variable and power.

> `3a + 5b − a + 2b = (3a − a) + (5b + 2b) = 2a + 7b`

• Only add or subtract terms with identical letter parts.
• Constants collect with constants.

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2. Multiplying and Dividing Terms (Index Laws)

When multiplying algebraic terms:

• Multiply the coefficients.
• Add the powers of the same base.

When dividing algebraic terms:

• Divide the coefficients.
• Subtract the powers of the same base.

| Operation | Rule | Example |

|-----------|------|---------|

| Multiplication | `aᵐ × aⁿ = aᵐ⁺ⁿ` | `3x² × 4x³ = 12x⁵` |

| Division | `aᵐ ÷ aⁿ = aᵐ⁻ⁿ` | `10x⁶ ÷ 2x² = 5x⁴` |

| Power of a power | `(aᵐ)ⁿ = aᵐⁿ` | `(2x³)² = 4x⁶` |

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3. Expanding Single Brackets

Multiply the term outside the bracket by every term inside:

> `a(b + c) = ab + ac`

• Pay careful attention to signs when a negative is outside the bracket.

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4. Expanding and Simplifying Double Brackets

Use the FOIL method (First, Outer, Inner, Last) or the grid method:

> `(x + a)(x + b) = x² + bx + ax + ab`

Then collect like terms.

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5. Expanding Triple Brackets (Higher Tier)

Expand two brackets first, then multiply the result by the third bracket.

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6. Substitution into Expressions

Replace each variable with its given numerical value and evaluate, respecting BIDMAS (order of operations).

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Worked Examples

Worked Example 1 – Simplifying and Expanding

Expand and simplify `3(2x + 5) − 2(x − 4)`

| Step | Working |

|------|---------|

| Expand first bracket | `3 × 2x + 3 × 5 = 6x + 15` |

| Expand second bracket | `−2 × x + (−2) × (−4) = −2x + 8` |

| Combine | `6x + 15 − 2x + 8` |

| Collect like terms | `4x + 23` |

Answer: `4x + 23` ✓

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Worked Example 2 – Expanding Double Brackets

Expand and simplify `(2x − 3)(x + 5)`

| Step | Working |

|------|---------|

| First | `2x × x = 2x²` |

| Outer | `2x × 5 = 10x` |

| Inner | `−3 × x = −3x` |

| Last | `−3 × 5 = −15` |

| Combine | `2x² + 10x − 3x − 15` |

| Simplify | `2x² + 7x − 15` |

Answer: `2x² + 7x − 15` ✓

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Worked Example 3 – Substitution

Given `P = 3a² − 2b`, find the value of *P* when `a = −4` and `b = 5`.

| Step | Working |

|------|---------|

| Substitute | `P = 3(−4)² − 2(5)` |

| Evaluate powers first | `P = 3(16) − 10` |

| Multiply | `P = 48 − 10` |

| Final answer | `P = 38` |

Answer: `P = 38` ✓

> *Note*: A common error is writing `(−4)² = −16`. Remember that squaring a negative gives a positive result.

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Exam Technique Tips

Tip 1 – Show Every Line of Working

Edexcel mark schemes award method marks (M marks) for correct expansion or simplification steps, even if your final answer is wrong. For example, when expanding double brackets, write out all four terms before collecting like terms. This protects you from losing all marks due to a small arithmetic slip.

Tip 2 – Watch for "Expand and Simplify"

When a question says *"expand and simplify"*, you must collect like terms after expanding to earn the final accuracy mark (A mark). Simply expanding without simplifying will cost you the last mark. Underline or circle the command words in the question to remind yourself of what is required.

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*End of revision notes – Algebraic Expressions*