Algebraic manipulation

1. Overview

Algebraic manipulation is the foundation for success in IGCSE Mathematics. It involves rewriting expressions through simplifying, expanding, and factorising. These skills are crucial for solving equations, working with functions, and tackling calculus problems. This revision guide covers the core techniques and extends to more advanced factorisation methods.

Key Definitions

• Term: A single number, variable, or numbers and variables multiplied together (e.g., $3$, $x$, or $5xy^2$).
• Expression: A group of terms linked by plus or minus signs (e.g., $2x + 3y$).
• Like Terms: Terms that have the exact same variables raised to the exact same powers (e.g., $4ab$ and $7ab$ are like terms; $4a^2$ and $4a$ are not).
• Expand: To remove brackets by multiplying the term outside by everything inside.
• Factorise: The inverse of expanding; writing an expression as a product of its factors using brackets.
• Coefficient: The number in front of a variable (e.g., in $5x^2$, the coefficient is $5$).

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Core Content

3.1 Simplifying Expressions (Collecting Like Terms)

To simplify algebraic expressions, combine "like terms" by adding or subtracting their coefficients. Remember, you can only combine terms with the same variable raised to the same power. Ensure you collect all like terms fully.

Worked example 1 — Simplifying an expression

Simplify: $7a - 4b + 2a + 5b - 3$

$7a - 4b + 2a + 5b - 3$ — Original expression $7a + 2a - 4b + 5b - 3$ — Rearrange to group like terms $(7 + 2)a + (-4 + 5)b - 3$ — Combine coefficients $9a + 1b - 3$ — Simplify $9a + b - 3$ — Final simplified expression

Worked Example 2 — Simplifying with exponents Simplify: $3x^2 + 4x - x^2 + 2x - 5$

$3x^2 + 4x - x^2 + 2x - 5$ — Original expression $3x^2 - x^2 + 4x + 2x - 5$ — Group like terms $(3 - 1)x^2 + (4 + 2)x - 5$ — Combine coefficients $2x^2 + 6x - 5$ — Simplify $2x^2 + 6x - 5$ — Final simplified expression

3.2 Expanding Products

Expanding involves removing brackets by multiplying the term outside the bracket by each term inside. Use the distributive property carefully, paying attention to signs. For double brackets, the FOIL method (First, Outer, Inner, Last) is a helpful technique.

Worked example 3 — Expanding a single bracket

Expand: $-3y(5y - 2)$

$-3y(5y - 2)$ — Original expression $(-3y \times 5y) + (-3y \times -2)$ — Multiply -3y by each term inside the bracket $-15y^2 + 6y$ — Simplify $-15y^2 + 6y$ — Final expanded expression

Worked example 4 — Expanding double brackets

Expand: $(2x - 1)(x + 4)$

$(2x - 1)(x + 4)$ — Original expression $(2x \times x) + (2x \times 4) + (-1 \times x) + (-1 \times 4)$ — Apply FOIL method $2x^2 + 8x - x - 4$ — Multiply each term $2x^2 + 7x - 4$ — Combine like terms $2x^2 + 7x - 4$ — Final expanded expression

3.3 Factorising (Common Factors)

Factorising is the reverse of expanding. Identify the Highest Common Factor (HCF) of all terms in the expression and place it outside a bracket. Divide each term by the HCF to determine the terms inside the bracket.

Worked example 5 — Factorising with a common factor

Factorise: $12a^3b - 18ab^2$

$12a^3b - 18ab^2$ — Original expression HCF of 12 and 18 is 6. HCF of $a^3$ and $a$ is $a$. HCF of $b$ and $b^2$ is $b$. HCF is $6ab$. $6ab(2a^2 - 3b)$ — Divide each term by the HCF and place inside the bracket $6ab(2a^2 - 3b)$ — Final factorised expression

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Extended Content (Extended Only)

4.1 Factorising by Grouping

This technique is used for expressions with four terms. Group the terms into pairs, factorise each pair separately, and then look for a common bracket.

Worked example 6 — Factorising by grouping

Factorise: $pq + 2p + 3q + 6$

$pq + 2p + 3q + 6$ — Original expression $(pq + 2p) + (3q + 6)$ — Group terms into pairs $p(q + 2) + 3(q + 2)$ — Factorise each pair $(q + 2)(p + 3)$ — Take out the common bracket $(q + 2)(p + 3)$ — Final factorised expression

4.2 Difference of Two Squares ($a^2 - b^2$)

The difference of two squares can be factorised using the rule: $a^2 - b^2 = (a + b)(a - b)$. Recognise perfect squares within the expression.

Worked example 7 — Difference of two squares

Factorise: $49x^2 - 64y^2$

$49x^2 - 64y^2$ — Original expression $(7x)^2 - (8y)^2$ — Recognise perfect squares $(7x + 8y)(7x - 8y)$ — Apply the difference of two squares rule $(7x + 8y)(7x - 8y)$ — Final factorised expression

4.3 Factorising Quadratics ($ax^2 + bx + c$)

For quadratics where $a = 1$ (i.e., $x^2 + bx + c$), find two numbers that multiply to $c$ and add to $b$. For quadratics where $a > 1$, use the "ac method" (grouping).

Worked example 8 — Factorising a quadratic ($a=1$)

Factorise: $x^2 + 8x + 15$

$x^2 + 8x + 15$ — Original expression Find two numbers that multiply to 15 and add to 8: 3 and 5. $(x + 3)(x + 5)$ — Write the factors $(x + 3)(x + 5)$ — Final factorised expression

Worked example 9 — Factorising a quadratic ($a>1$)

Factorise: $3x^2 - 10x + 8$

$3x^2 - 10x + 8$ — Original expression Multiply $a \times c = 3 \times 8 = 24$. Find two numbers that multiply to 24 and add to -10: -6 and -4. $3x^2 - 6x - 4x + 8$ — Split the middle term $3x(x - 2) - 4(x - 2)$ — Factorise by grouping $(x - 2)(3x - 4)$ — Take out the common bracket $(x - 2)(3x - 4)$ — Final factorised expression

4.4 Factorising Cubics (Common Factor first)

Always look for a common factor first. After extracting the common factor, you may be left with a quadratic expression that can be further factorised.

Worked example 10 — Factorising a cubic

Factorise: $5x^3 + 10x^2 - 15x$

$5x^3 + 10x^2 - 15x$ — Original expression $5x(x^2 + 2x - 3)$ — Take out the HCF 5x Now factorise the quadratic $x^2 + 2x - 3$. Find two numbers that multiply to -3 and add to 2: 3 and -1. $5x(x + 3)(x - 1)$ — Factorise the quadratic $5x(x + 3)(x - 1)$ — Final factorised expression

4.5 Completing the Square

Completing the square involves rewriting a quadratic expression in the form $a(x + p)^2 + q$. This is particularly useful for finding the minimum or maximum value of a quadratic function.

Worked example 11 — Completing the square ($a=1$)

Complete the square for: $x^2 - 8x + 7$

$x^2 - 8x + 7$ — Original expression Halve the coefficient of $x$: $-8 \div 2 = -4$. Write in the form $(x - 4)^2$. Expand $(x - 4)^2 = x^2 - 8x + 16$. We need $7$, but we have $16$. Subtract $9$ ($7-16 = -9$). $(x - 4)^2 - 9$ — Final completed square form

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Key Equations

Note: These formulas are not provided on the IGCSE formula sheet; they must be memorised.

$(a + b)^2 = a^2 + 2ab + b^2$ — Perfect Square (Addition) $(a - b)^2 = a^2 - 2ab + b^2$ — Perfect Square (Subtraction) $a^2 - b^2 = (a + b)(a - b)$ — Difference of Two Squares $x^2 + bx = (x + \frac{b}{2})^2 - (\frac{b}{2})^2$ — Completing the Square ($a=1$)

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Common Mistakes to Avoid

• ❌ Wrong: $5x - 2x + y = 3x^2 + y$ ✓ Right: $5x - 2x + y = 3x + y$. (Only combine terms with the same variable and power.)
• ❌ Wrong: Expanding $2(x - 3)$ gives $2x - 3$ ✓ Right: Expanding $2(x - 3)$ gives $2x - 6$. (Multiply every term inside the bracket by the term outside.)
• ❌ Wrong: Factorising $4x^2 + 8x$ as $2x(2x + 4)$. ✓ Right: Factorising $4x^2 + 8x$ as $4x(x + 2)$. (Always extract the highest common factor.)
• ❌ Wrong: $(x - 3)^2 = x^2 - 9$ ✓ Right: $(x - 3)^2 = (x - 3)(x - 3) = x^2 - 6x + 9$. (Remember the middle term when squaring a bracket.)
• ❌ Wrong: $x^2 + 4$ can be factorised as $(x+2)(x-2)$ ✓ Right: $x^2 + 4$ cannot be factorised using real numbers. $(x+2)(x-2) = x^2 - 4$.

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Exam Tips

• Command Words:
  • "Factorise" means you must include brackets in your answer.
  • "Factorise completely" indicates multiple steps are needed (e.g., common factor followed by quadratic factorisation).
• Check Your Work: After factorising, expand your answer to verify it matches the original expression.
• Non-Calculator Paper: Strong times tables knowledge speeds up factor identification in quadratics.
• The "Hidden" 1: Remember $x = 1x$, useful for collecting terms like $5x - x = 4x$.
• Sign Awareness: $x^2 - 4$ (difference of squares) factorises, but $x^2 + 4$ does not (using real numbers).
• Fully Simplified: Ensure all terms are in their simplest form before combining like terms. For example, simplify $2 \times 3 \times x$ to $6x$ before attempting to collect terms.