Summary of Fundamental Algebraic Operations

Fundamental Algebraic Operations: Guide for Students

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Introduction

Algebra is the language of patterns and relationships. In this guide we focus on expressions and factorisation, two core skills used to simplify, manipulate, and solve algebraic problems. You will learn how to collect like terms, expand and distribute brackets, and factor expressions by extracting the highest common factor (HCF). Clear worked examples and short exercises will help you practise.

Definition: An algebraic expression is a combination of numbers, variables and operations. Factorisation is writing an expression as a product of simpler expressions.

1. Collecting like terms

Like terms have the same variable parts (including powers). Combine them by adding or subtracting their coefficients.

  • Step 1: Identify like terms
  • Step 2: Add or subtract coefficients
  • Step 3: Keep unlike terms separate

Examples:

$$-12 + 8y - 2x + 6x + 9 + 6y - 4$$ Combine like terms by grouping $x$-terms, $y$-terms and constants:

$$-2x + 6x = 4x$$ $$8y + 6y = 14y$$ $$-12 + 9 - 4 = -7$$ Hence the simplified result is:

$$4x + 14y - 7$$

Note: The original scribble produced $4x + 14y - 12$ but correct arithmetic on the constants gives $-7$.

Another worked grouping example:

$$-2x + 6x + 8y + 6y - 12 + 9 - 9$$ Combine:

$$-2x + 6x = 4x$$ $$8y + 6y = 14y$$ $$-12 + 9 - 9 = -12$$ So:

$$4x + 14y - 12$$

2. Distributive property (expanding brackets)

The distributive property states that you multiply every term inside a bracket by the factor outside:

Definition: For all numbers or expressions $a$, $b$, $c$ we have $a\left(b + c\right) = ab + ac$.

Examples:

$$8(xy + 3x - 4y - 3) - 4(2 - 4 - 5x)$$ Apply distribution to each bracket separately:

$$8(xy) + 8(3x) - 8(4y) - 8(3) - 4(2) + 4(4) + 4(5x)$$ Simplify coefficients:

$$8xy + 24x - 32y - 24 - 8 + 16 + 20x$$ Combine like terms (constants and $x$-terms):

$$24x + 20x = 44x$$ $$-32y$$ stays as is $$-24 - 8 + 16 = -16$$ So the simplified expression is:

$$8xy + 44x - 32y - 16$$

Tip: Always expand each bracket carefully and then combine like terms.

💡 Did you know?Fun fact: The distributive property is the key step that connects multiplication and addition and is essential for simplifying algebraic expressions.

3. Common mistakes when expanding

  • Forgetting to multiply every term inside a bracket
  • Sign errors when subtracting a bracketed expression
  • Arithmetic mistakes when combining constants

Example of a sign trap:

$$-4(2 - 4 - 5x)$$ Distribute the $-4$ to each term:

$$-8 + 16 + 20x$$

4. Factorisation by taking out the Highest Common Factor (HCF)

Factorisation often means extracting the greatest factor common to all terms.

Definition: The highest common factor (HCF) of terms is the largest expression that divides each term without remainder.

Steps:

  1. Identify the numerical HCF of coefficients
  2. Identify common variable factors and their lowest powers
  3. Factor them out and write the remaining expression inside brackets

Examples:

$$2x + 10 = 2\left(x + 5\right)$$

$$8 - 12y = 4\left(2 - 3y\right)$$

$$6xy + 12y = 6y\left(x + 2\right)$$

Careful example with symbols:

$$4a + 8ab = 4a\left(1 + 2b\right)$$

Note: The scribble had a line "$$\pi^2 - 5x = \pi(x + 5)$$" which is incorrect. You cannot factor $\pi^2 - 5x$ as $\pi(x + 5)$ because the terms are not multiples of $\pi$. Correct factorisation requires a common factor present in every term.

💡 Did you know?Did you know that factoring expressions is useful for simplifying algebraic fractions and solving polynomial equations by finding roots?

5. Factorising quadratic-like expressions (simple cases)

When terms share a common factor:

$$4x^2 + 5x$$ We can factor an $x$ (if present) but here the scribbled factorisation $\pi(4x + 5)$ is wrong unless $x$ contains $\pi$. The correct factorisation is:

$$x\left(4x + 5\right)$$

Another example:

$$6xy + 12y = 6y\left(x + 2\right)$$

6. Solving simple linear equations (using collected terms and factorisation)

General steps:

  1. Collect like terms on each side
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Algebra: Expressions & Factorisation

Klíčové pojmy: Combine like terms by adding/subtracting coefficients of identical variable parts, Distribute multiplication over addition: $a\left(b+c\right)=ab+ac$, When subtracting a bracket, multiply each term by the negative sign, Factorisation: extract the highest common factor (HCF) from all terms, Check factorisation by expanding the factored form, Solve linear equations by collecting terms, isolating the variable, then dividing, Always recheck arithmetic on constants separately to avoid mistakes, For expressions like $4x^2+5x$, factor out the common $x$ to get $x\left(4x+5\right)$, Do not assume symbolic constants (like $\pi$) are common factors unless they appear in every term, Use step-by-step writing: expand, combine like terms, then simplify

## Introduction Algebra is the language of patterns and relationships. In this guide we focus on **expressions** and **factorisation**, two core skills used to simplify, manipulate, and solve algebraic problems. You will learn how to collect like terms, expand and distribute brackets, and factor expressions by extracting the highest common factor (HCF). Clear worked examples and short exercises will help you practise. > Definition: An algebraic expression is a combination of numbers, variables and operations. Factorisation is writing an expression as a product of simpler expressions. ## 1. Collecting like terms Like terms have the same variable parts (including powers). Combine them by adding or subtracting their coefficients. - Step 1: Identify like terms - Step 2: Add or subtract coefficients - Step 3: Keep unlike terms separate Examples: $$-12 + 8y - 2x + 6x + 9 + 6y - 4$$ Combine like terms by grouping $x$-terms, $y$-terms and constants: $$-2x + 6x = 4x$$ $$8y + 6y = 14y$$ $$-12 + 9 - 4 = -7$$ Hence the simplified result is: $$4x + 14y - 7$$ Note: The original scribble produced $4x + 14y - 12$ but correct arithmetic on the constants gives $-7$. Another worked grouping example: $$-2x + 6x + 8y + 6y - 12 + 9 - 9$$ Combine: $$-2x + 6x = 4x$$ $$8y + 6y = 14y$$ $$-12 + 9 - 9 = -12$$ So: $$4x + 14y - 12$$ ## 2. Distributive property (expanding brackets) The distributive property states that you multiply every term inside a bracket by the factor outside: > Definition: For all numbers or expressions $a$, $b$, $c$ we have $a\left(b + c\right) = ab + ac$. Examples: $$8(xy + 3x - 4y - 3) - 4(2 - 4 - 5x)$$ Apply distribution to each bracket separately: $$8(xy) + 8(3x) - 8(4y) - 8(3) - 4(2) + 4(4) + 4(5x)$$ Simplify coefficients: $$8xy + 24x - 32y - 24 - 8 + 16 + 20x$$ Combine like terms (constants and $x$-terms): $$24x + 20x = 44x$$ $$-32y$$ stays as is $$-24 - 8 + 16 = -16$$ So the simplified expression is: $$8xy + 44x - 32y - 16$$ Tip: Always expand each bracket carefully and then combine like terms. Fun fact: The distributive property is the key step that connects multiplication and addition and is essential for simplifying algebraic expressions. ## 3. Common mistakes when expanding - Forgetting to multiply every term inside a bracket - Sign errors when subtracting a bracketed expression - Arithmetic mistakes when combining constants Example of a sign trap: $$-4(2 - 4 - 5x)$$ Distribute the $-4$ to each term: $$-8 + 16 + 20x$$ ## 4. Factorisation by taking out the Highest Common Factor (HCF) Factorisation often means extracting the greatest factor common to all terms. > Definition: The highest common factor (HCF) of terms is the largest expression that divides each term without remainder. Steps: 1. Identify the numerical HCF of coefficients 2. Identify common variable factors and their lowest powers 3. Factor them out and write the remaining expression inside brackets Examples: $$2x + 10 = 2\left(x + 5\right)$$ $$8 - 12y = 4\left(2 - 3y\right)$$ $$6xy + 12y = 6y\left(x + 2\right)$$ Careful example with symbols: $$4a + 8ab = 4a\left(1 + 2b\right)$$ Note: The scribble had a line "$$\pi^2 - 5x = \pi(x + 5)$$" which is incorrect. You cannot factor $\pi^2 - 5x$ as $\pi(x + 5)$ because the terms are not multiples of $\pi$. Correct factorisation requires a common factor present in every term. Did you know that factoring expressions is useful for simplifying algebraic fractions and solving polynomial equations by finding roots? ## 5. Factorising quadratic-like expressions (simple cases) When terms share a common factor: $$4x^2 + 5x$$ We can factor an $x$ (if present) but here the scribbled factorisation $\pi(4x + 5)$ is wrong unless $x$ contains $\pi$. The correct factorisation is: $$x\left(4x + 5\right)$$ Another example: $$6xy + 12y = 6y\left(x + 2\right)$$ ## 6. Solving simple linear equations (using collected terms and factorisation) General steps: 1. Collect like terms on each side 2.