Factorisation – Part Two: Trinomial Expansion

Meaning of a Quadratic Expression

A quadratic expression is an algebraic expression in which the highest power of the variable is 2.
Examples:

1. x² + 5x + 6,
2. y² − 3y + 2,
3. 2a² + 7a + 3

Notice that in each case the largest power of the variable is 2.

When a quadratic expression has three terms, we call it a trinomial.

Link Between Expansion and Factorisation

Before learning to factorise, it helps to see how these expressions are formed.

Expansion means multiplying out two brackets.

Example:

(w + 2)(w + 3)

= w(w + 3) + 2(w + 3) (break one bracket [w] and [+2] and multiply each broken term by the other unbroken bracket  (w+3))

= w² + 3w + 2w + 6

= w² + 5w + 6

So the trinomial w² + 5w + 6 comes from multiplying two binomials.

Factorisation is simply the reverse process:

w² + 5w + 6 ⟶ (w + 2)(w + 3)

Step–by–Step: Expanding Two Brackets

Multiply the first terms: w × w = w²

Multiply the outer terms: w × (+ 3) = +3w

Multiply the inner terms: (+2) × w = +2w

Multiply the last terms: (+2) × (+3) = +6

Simplify: w² + 3w + 2w + 6 = w² + 5w + 6

This is called the double distributive method or FOIL (First, Outer, Inner, Last).

More Examples of Expansion

Example 1

(x + 4)(x + 1)

= x(x + 1) + 4(x + 1)

= x² + x + 4x + 4

= x² + 5x + 4

Example 2

(x − 3)(x + 2)

= x(x + 2) - 3(x + 2)

= x² + 2x − 3x − 6

= x² − x − 6

Example 3

(x − 5)(x − 1)

= x(x - 1)  - 5(x - 1)

= x² − x − 5x + 5

= x² − 6x + 5

 Quick Practice

Expand these:

1. (x + 1)(x + 2)

2. (x + 3)(x − 4)

3. (x − 2)(x − 5)

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