Algebraic Multiplication – Expanding Brackets
Multiplying in algebra means applying the distributive law to expand brackets and then simplifying by collecting like terms.
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 Questions
- Expand 4(3x − 2).
- Expand (x + 6)(x + 2).
- Expand (x − 5)(x + 3).
- Expand (2x + 1)(3x − 4).
- Expand (x + 3)².