Algebraic Multiplication – Expanding Brackets

Multiplying in algebra means applying the distributive law to expand brackets and then simplifying by collecting like terms.

The perfect-square pattern (x + a)² = x² + 2ax + a² is one of the most frequently reused identities in algebra, appearing everywhere from completing the square when solving quadratics to the geometric proof of the Pythagorean theorem, where the area of a square with side (a+b) is shown to split into four regions matching exactly this expansion. Recognising this pattern on sight — rather than re-deriving it with FOIL every time — is a genuine time-saver once you reach more advanced algebra.

Multiplying a Single Term by a Bracket

3x(2x + 5) = (3x)(2x) + (3x)(5) = 6x² + 15x

Multiplying Two Binomials – The FOIL Method

FOIL stands for: First, Outer, Inner, Last.

(x + 3)(x + 5)

First: x × x = x².

Outer: x × 5 = 5x.

Inner: 3 × x = 3x.

Last: 3 × 5 = 15.

Combine: x² + 5x + 3x + 15 = x² + 8x + 15

(2x − 3)(x + 4)

2x² + 8x − 3x − 12 = 2x² + 5x − 12

Expanding a Perfect Square

(x + 4)² = (x + 4)(x + 4) = x² + 8x + 16

Multiplying Polynomials

(x + 2)(x² − 3x + 1) = x³ − 3x² + x + 2x² − 6x + 2 = x³ − x² − 5x + 2

Key Takeaways

Every term in the first bracket multiplies every term in the second.
FOIL is a useful reminder for two binomials specifically.
Always collect and simplify like terms after expanding.
Multiply coefficients and add exponents of the same variable.

Practice: Algebraic Multiplication

Multiply a Single Term by a Bracket