Polynomials
A polynomial is an algebraic expression consisting of variables and coefficients combined using addition, subtraction, and multiplication, where variables have non-negative integer powers. The term 'polynomial' comes from Greek, meaning 'many terms'. Common examples include 3x + 5 (linear), x² - 2x + 1 (quadratic), and 2x³ + x² - 4x + 7 (cubic).
Why it matters
Polynomials form the foundation of algebra and appear throughout GCSE mathematics, particularly in quadratic equations and graph sketching. Engineers use polynomial functions to model curved relationships, such as the trajectory of a football kick reaching a maximum height of 15 metres. Economic models rely on polynomials to predict profit margins, where a company might use P(x) = -2x² + 100x - 800 to determine optimal production levels. In physics, polynomial equations describe motion under gravity, where distance equals ½at² + vt + s. Computer graphics use polynomial curves to create smooth animations and 3D models. Architecture employs polynomial functions to design parabolic arches that can span 50-metre bridges whilst maintaining structural integrity.
How to solve polynomials
Polynomials
- To add/subtract: combine like terms (same power of x).
- To multiply: use FOIL or distribute each term.
- To factor: find two numbers that multiply to c and add to b.
Example: (x+2)(x+3) = x² + 5x + 6.
Worked examples
(2x + 1) + (1x + 4) = _______
Answer: 3x + 5
- Combine like terms → 2x + 1x = 3x, 1 + 4 = 5 — Add x-terms together and constants together.
- Write result → 3x + 5 — Combined polynomial.
(2x − 2) + (3x − 5) = _______
Answer: 5x − 7
- Combine like terms → 5x − 7 — + the x-terms and constants separately.
(3x − 3)(1x − 1) = _______
Answer: 3x² + -6x + 3
- FOIL: First → 3x · 1x = 3x² — Multiply the first terms.
- Outer + Inner → 3x·-1 + -3·1x = -3x + -3x = -6x — Multiply outer and inner, combine.
- Last → -3 · -1 = 3 — Multiply the last terms.
- Combine → 3x² + -6x + 3 — Write the expanded polynomial.
Common mistakes
- When adding polynomials, combining unlike terms incorrectly, such as writing 2x² + 3x = 5x³ instead of leaving them separate as 2x² + 3x.
- During FOIL multiplication, forgetting the middle terms and writing (x + 2)(x + 3) = x² + 6 instead of x² + 5x + 6.
- When factoring x² + 7x + 12, choosing factors that multiply to 12 but don't add to 7, such as 2 and 6, rather than the correct 3 and 4.