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Removing Brackets

Everything inside the bracket must be multiplied by everything outside the bracket.

Examples

Single Brackets

\[ 3(x + 4) = 3x + 12 \] \[ 5(2x - 7) = 10x - 35 \] \[ -2(3x + 5) = -6x - 10 \] \[ 4(x - 9) = 4x - 36 \]

Pairs of Brackets

Example

Expand:

\[ (x+2)(x-4) \]

Use distributive law:

\[ (x+2)(x-4) = x(x-4) + 2(x-4) \]

Expand each part:

\[ x(x-4) = x^2 - 4x \] \[ 2(x-4) = 2x - 8 \]

Combine and simplify:

\[ x^2 - 4x + 2x - 8 = x^2 - 2x - 8 \]
General expansion:
\[ (a + b)(c + d) = ac + ad + bc + bd \]

Is there an easier way?

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foil mnemonic
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Example
\[ (x+3)(x+5) \] \[ \text{FOIL: First, Outer, Inner, Last} \] \[ = x\cdot x \;+\; x\cdot 5 \;+\; 3\cdot x \;+\; 3\cdot 5 \] \[ = x^2 + 5x + 3x + 15 \] \[ = x^2 + 8x + 15 \]
Example
\[ (2x - 7)(x + 4) \] \[ \text{FOIL: First, Outer, Inner, Last} \] \[ = (2x)(x) \;+\; (2x)(4) \;+\; (-7)(x) \;+\; (-7)(4) \] \[ = 2x^2 + 8x - 7x - 28 \] \[ = 2x^2 + x - 28 \]

Trinomials

A trinomial is an algebraic expression made up of three terms, usually involving powers of a variable. When multiplying or expanding expressions, trinomials appear naturally — especially when you multiply a binomial by another binomial or when you expand something like or . To expand a trinomial, you multiply each term carefully and then collect like terms.

FOIL‑Plus Method

FOIL works for two terms × two terms.
FOIL‑plus extends the idea: Multiply each term in the first bracket by the entire second bracket.

Example

\[ (x + 2)(x^2 + 3x + 4) \]

Step 1: Multiply each term in the binomial across the trinomial

\[ = x(x^2 + 3x + 4) + 2(x^2 + 3x + 4) \]

Step 2: Expand each part

\[ = x^3 + 3x^2 + 4x + 2x^2 + 6x + 8 \]

Step 3: Collect like terms

\[ = x^3 + 5x^2 + 10x + 8 \]

FOIL‑Plus Summary:
Multiply → Expand → Collect

Example

Expand \( (x+2)(3x+2)^2 \)

\[ (x+2)(3x+2)^2 \] \[ = (x+2)(3x+2)(3x+2) \] \[ = (x+2)(3x^2 + 6x + 6x + 4) \] \[ = (x+2)(3x^2 + 12x + 4) \] \[ = x \times 3x^2 + x \times 12x + x \times 4 + 2 \times 3x^2 + 2 \times 12x + 2 \times 4 \] \[ = 3x^3 + 12x^2 + 4x + 6x^2 + 24x + 8 \] \[ = 3x^3 + 18x^2 + 28x + 8 \]

General Result

Expanding \((a+b)^2\)

\[ (a+b)^2 = (a+b)(a+b) \] \[ = a\cdot a + a\cdot b + b\cdot a + b\cdot b \] \[ = a^2 + 2ab + b^2 \]

Pattern: Square – Double – Square

Expanding \((a-b)^2\)

\[ (a-b)^2 = (a-b)(a-b) \] \[ = a\cdot a - a\cdot b - b\cdot a + b\cdot b \] \[ = a^2 - 2ab + b^2 \]

Pattern: Square – Double (negative) – Square

Expanding \((a+b)(a-b)\)

\[ (a+b)(a-b) \] \[ = a\cdot a - a\cdot b + b\cdot a - b\cdot b \] \[ = a^2 - b^2 \]

Pattern: Difference of two squares

Expanding \((a+b)^3\)

\[ (a+b)^3 = (a+b)(a+b)(a+b) \] \[ = a^3 + 3a^2b + 3ab^2 + b^3 \]

Pattern: Pascal’s triangle row 3 → 1, 3, 3, 1

Expanding \((a-b)^3\)

\[ (a-b)^3 = (a-b)(a-b)(a-b) \] \[ = a^3 - 3a^2b + 3ab^2 - b^3 \]

Pattern: Signs alternate: +, −, +, −

General Binomial Expansion: \((a+b)^n\)

Binomial Theorem:

\[ (a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{\,n-k} b^{\,k} \]

Coefficients: Pascal’s triangle row \(n\) Binomial Expansion

General Binomial Expansion: \((a-b)^n\)

Binomial Theorem with alternating signs:

\[ (a-b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{\,n-k} (-b)^{\,k} \]

Sign rule: Terms alternate depending on \(k\)

Factorising

(Bringing back brackets)

Find the HCF and place it outside the brackets.

Examples
\[ 12x + 18 = 6(2x + 3) \] \[ 15x^2 - 10x = 5x(3x - 2) \] \[ 21x^3 + 14x^2 = 7x^2(3x + 2) \]

The Difference of Two Squares

\[ a^2 - b^2 = (a - b)(a + b) \]
Examples
\[ x^2 - 25 = (x - 5)(x + 5) \] \[ 4x^2 - 49 = (2x - 7)(2x + 7) \] \[ 9a^2 - 16b^2 = (3a - 4b)(3a + 4b) \] \[ 25x^2 - y^2 = (5x - y)(5x + y) \]

When factorising

 

  1. Look for common factors
  2. Look for difference of two squares
  3. Look to factorise quadratics
  4. Use the Quadratic Formula
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