Algebraic Manipulation

Single bracket. $a(b + c) = ab + ac$. Multiply $a$ by each term inside.

Example: $3x(2x - 5) = 6x^{2} - 15x$.

Double brackets — FOIL. $(a + b)(c + d) = ac + ad + bc + bd$.

Step	Description
F	First × First
O	Outer × Outer
I	Inner × Inner
L	Last × Last

Example: $(2x + 3)(x - 4)$

• F: $2x \cdot x = 2x^{2}$.
• O: $2x \cdot (-4) = -8x$.
• I: $3 \cdot x = 3x$.
• L: $3 \cdot (-4) = -12$.
• Combine: $2x^{2} - 8x + 3x - 12 = 2x^{2} - 5x - 12$.

Three brackets. Do TWO first, then expand by the third.

Example: $(x + 1)(x - 2)(x + 3)$

• $(x+1)(x-2) = x^{2} - x - 2$.
• $(x^{2} - x - 2)(x + 3) = x^{3} + 3x^{2} - x^{2} - 3x - 2x - 6$.
• Simplify: $x^{3} + 2x^{2} - 5x - 6$.

Squaring binomials — TWO patterns to memorise:

$\boxed{(a + b)^{2} = a^{2} + 2ab + b^{2}}$ $\boxed{(a - b)^{2} = a^{2} - 2ab + b^{2}}$

Both have THREE terms. Forgetting the middle is a classic A* trap.

Difference of squares:

$\boxed{(a - b)(a + b) = a^{2} - b^{2}}$

Two terms — the cross-terms cancel.

Worked qualitative. Why does $(a + b)(a - b)$ give only two terms?

• FOIL: $a^{2} - ab + ab - b^{2}$.
• Middle terms $-ab + ab = 0$.
• Result: $a^{2} - b^{2}$ — two terms.
• Useful for factoring questions later.

Edexcel tip. Show every FOIL step. Mark schemes give M1 for any 3 of 4 terms correct, A1 for the simplified answer.

Like terms have the same variable raised to the same power.

Like?	Why
$3x$ and $5x$	✓ Same variable, same power
$4x^{2}$ and $-7x^{2}$	✓ Same
$2x$ and $2x^{2}$	✗ Different powers
$3x$ and $3y$	✗ Different variables
$5xy$ and $-2xy$	✓ Same

Collecting: add the coefficients of like terms.

Example: $5x^{2} - 3x + 7 - 2x^{2} + 5x - 4$.

Group:

• $x^{2}$: $5x^{2} - 2x^{2} = 3x^{2}$.
• $x$: $-3x + 5x = 2x$.
• Constants: $7 - 4 = 3$.

Result: $3x^{2} + 2x + 3$.

Algebra fluency rules:

• $x \cdot x = x^{2}$.
• $x \cdot 0 = 0$.
• $0 \cdot x = 0$.
• $1 \cdot x = x$ (drop the $1$).
• $-1 \cdot x = -x$.
• $x + x = 2x$ (NOT $x^{2}$).
• $x \cdot x = x^{2}$ (NOT $2x$).

Worked qualitative. Why does $x + x = 2x$ but $x \cdot x = x^{2}$?

• $x + x$: adding two of the same — count them. $2x$.
• $x \cdot x$: multiplying two of the same — multiply means power up. $x^{2}$.
• Different operations, different rules.

Edexcel tip. When simplifying multi-power expressions, write powers DESCENDING (highest first): $3x^{2} + 2x + 3$. Mark schemes prefer this convention.

Expression. A combination of numbers, variables, operations — NO equals sign.

Examples: $3x^{2} + 2x - 5$, $\dfrac{x+1}{x-3}$, $\sqrt{2x+1}$.

You can SIMPLIFY an expression but not 'solve' it.

Equation. A statement: two expressions are equal, using $=$. True for SOME values of the variable (the solutions).

Examples: $3x + 2 = 11$ (solution $x = 3$). $x^{2} = 16$ (solutions $x = \pm 4$).

You SOLVE equations.

Identity. A statement that is TRUE for ALL values of the variable. Often written with $\equiv$.

Examples:

• $(a + b)^{2} \equiv a^{2} + 2ab + b^{2}$ (always true).
• $\sin^{2}\theta + \cos^{2}\theta \equiv 1$ (always true).
• $a(b + c) \equiv ab + ac$ (distributive identity).

Worked qualitative. Is $x^{2} = 16$ an equation or identity?

• True only for $x = 4$ or $x = -4$, not for ALL $x$.
• It's an EQUATION.

Worked qualitative. Is $(a + b)^{2} = a^{2} + b^{2}$ an identity?

• $(1 + 2)^{2} = 9$, but $1^{2} + 2^{2} = 5$. Different.
• NOT true for all values.
• NOT an identity. (And, as a matter of fact, it's never true except when $a = 0$ or $b = 0$.)

Edexcel tip. Sometimes Edexcel asks 'Show that ... is an identity'. You need to prove it works for ALL values, usually by expanding both sides until they're equal.