Operations on Algebraic Expressions
Algebraic expressions are combinations of variables, constants, and mathematical operations. When we work with polynomials — expressions such as 3x² + 2x − 5 — we need precise rules for combining them through addition, subtraction, multiplication, and division. Mastering these four operations forms the backbone of all higher algebra, from solving equations to analysing functions. The key idea running through addition and subtraction is that only like terms — terms sharing exactly the same variable part — can be combined. For multiplication, we apply the distributive property systematically, including the important FOIL strategy for binomials. Division of polynomials introduces long division or synthetic division, methods that mirror the arithmetic long division you already know but applied to expressions in x. These operations are not merely symbol manipulation. They model real situations: multiplying two linear expressions can represent the area of a rectangle whose sides are described by those expressions, while dividing polynomials arises naturally when factoring or simplifying rational expressions. Understanding the structure behind each operation gives you both fluency and insight.
- Addition and Subtraction: Combining Like Terms
- Multiplication: Distributive Property and FOIL
- Division: Polynomial Long Division
- Special Products Worth Memorising
Addition and Subtraction: Combining Like Terms
Two terms are like terms if they have identical variable factors with identical exponents. For example, 4x²y and −9x²y are like terms, but 4x²y and 4xy² are not. To add or subtract polynomials, group like terms and add or subtract their coefficients: (3x² + 2x − 1) + (x² − 5x + 4) = (3+1)x² + (2−5)x + (−1+4) = 4x² − 3x + 3. For subtraction, distribute the negative sign first: (3x² + 2x − 1) − (x² − 5x + 4) = 3x² + 2x − 1 − x² + 5x − 4 = 2x² + 7x − 5.
Multiplication: Distributive Property and FOIL
To multiply a monomial by a polynomial, multiply the monomial by each term separately: 3x(2x² − x + 4) = 6x³ − 3x² + 12x. To multiply two binomials, use FOIL — First, Outer, Inner, Last: (x + 3)(x − 2) = x·x + x·(−2) + 3·x + 3·(−2) = x² − 2x + 3x − 6 = x² + x − 6. For larger polynomials, distribute each term of the first polynomial across every term of the second, then collect like terms. The degree of the product equals the sum of the degrees of the factors.
Division: Polynomial Long Division
To divide a polynomial p(x) by a divisor d(x), use the same algorithm as numeric long division. Write the dividend in standard form (descending powers, inserting 0 as placeholder for missing degrees). Divide the leading term of the dividend by the leading term of the divisor to get the first term of the quotient. Multiply that term by the entire divisor, subtract from the dividend, bring down the next term, and repeat. The process ends when the remainder has degree less than the degree of the divisor. The result is expressed as: p(x) ÷ d(x) = q(x) + r(x)/d(x), where r(x) is the remainder.
Special Products Worth Memorising
Several multiplication patterns appear so frequently that memorising them saves significant time. The perfect square trinomials: (a + b)² = a² + 2ab + b² and (a − b)² = a² − 2ab + b². The difference of squares: (a + b)(a − b) = a² − b². The sum/difference of cubes: (a + b)(a² − ab + b²) = a³ + b³ and (a − b)(a² + ab + b²) = a³ − b³. Recognising these patterns lets you multiply or factor expressions quickly and reliably.
The Key Equation
- p(x) — Dividend (—): The polynomial being divided; must be written in standard (descending) form before dividing.
- d(x) — Divisor (—): The polynomial you are dividing by; its leading term drives each step of the long-division algorithm.
- q(x) — Quotient (—): The polynomial result of the division; its degree equals deg(p) − deg(d).
- r(x) — Remainder (—): What is left over after division; its degree is always less than the degree of d(x). If r(x) = 0, then d(x) is a factor of p(x).
Worked Example
Problem. Divide p(x) = 2x³ − 3x² + x + 5 by d(x) = x − 2 using polynomial long division.
Answer. 2x² + x + 3 + 11/(x − 2)
Origins · 1 minFrançois Viète and how this was discovered
Viète introduced the use of letters to represent both known and unknown quantities systematically, transforming algebra from a collection of numerical recipes into a general symbolic language. His 1591 work In Artem Analyticem Isagoge established conventions that made algebraic operations — including the multiplication and manipulation of polynomial expressions — expressible in a compact, general form. Before Viète, algebra problems were described mostly in words or with abbreviations; his notation allowed operations on expressions to be performed and communicated universally. Viète worked as a lawyer and royal advisor to King Henry IV of France, and he famously cracked a Spanish military cipher — a code with 500 characters — that the Spanish king had claimed was unbreakable, reportedly attributing the French success to sorcery.
Related Concepts
- Factoring Polynomials — Factoring reverses multiplication of algebraic expressions; recognising products like (a+b)(a−b) lets you factor differences of squares instantly.
- Degree and Standard Form of Polynomials — Before performing addition, subtraction, or long division, polynomials must be written in standard form with terms ordered by descending degree.
- Distributive Property — The distributive property a(b+c) = ab + ac is the fundamental rule driving polynomial multiplication, including FOIL and expanding larger products.
- Rational Expressions — Dividing polynomials produces rational expressions p(x)/d(x); simplifying these requires polynomial long division and an understanding of remainders.