Algebraic Vocabulary
Algebraic Vocabulary
Precise language is essential in algebra. The words expression, equation, formula, identity and inequality each describe a distinct type of algebraic statement, and confusing them leads to fundamental errors — for example, trying to "solve" an identity, or not realising that an equation has specific solutions. This benchmark establishes the vocabulary used throughout all further algebra.
Five Types of Algebraic Statement
| Type | Symbol | Meaning | Example |
|---|---|---|---|
| Expression | — | A collection of terms. No equals sign. Cannot be "solved". | \( 3x^2 - 2x + 1 \) |
| Equation | \( = \) | States two expressions are equal. True only for specific values. | \( 3x + 2 = 11 \) (solved: \( x = 3 \)) |
| Formula | \( = \) | A rule relating two or more variables. Rearranged, not "solved". | \( A = \pi r^2 \) |
| Identity | \( \equiv \) | True for all values of the variable. Cannot be "solved". | \( 3(x+1) \equiv 3x+3 \) |
| Inequality | \( <, >, \leq, \geq \) | Describes a range of values. Solved to find the set of solutions. | \( 2x - 1 > 5 \) (\( x > 3 \)) |
The Identity Symbol ≡
An identity is different from an equation. The statement \( x^2 - 1 \equiv (x+1)(x-1) \) is true for every real number \( x \) — it is a structural algebraic truth, not something to solve. The equals sign in an equation asserts equality for specific values; the identity symbol \( \equiv \) asserts equality for all values. In practice, \( = \) is often used informally even for identities, but \( \equiv \) signals that no solutions need to be found.
Terms, Coefficients and Degree
In the expression \( 5x^3 - 3x^2 + 7x - 4 \):
- Terms: \( 5x^3 \), \( -3x^2 \), \( 7x \) and \( -4 \) — individual components separated by \( + \) or \( - \)
- Coefficients: 5 (of \( x^3 \)), \( -3 \) (of \( x^2 \)), 7 (of \( x \)) — the numerical multiplier of each term
- Constant term: \( -4 \) — contains no variable
- Degree: 3 — the highest power of the variable in the expression (so this is a cubic expression)
Common degree names: degree 1 = linear, degree 2 = quadratic, degree 3 = cubic.
Worked Examples
Classify each of the following:
\( 4x + 3 = 11 \) — Equation: contains \( = \); solutions exist (\( x = 2 \)).
\( A = lb \) — Formula: rule relating area, length and breadth.
\( 3x^2 - 2x + 1 \) — Expression: no equals sign.
\( 5x - 2 > 8 \) — Inequality: contains \( > \); solved as a range (\( x > 2 \)).
\( 2(x+3) \equiv 2x + 6 \) — Identity: true for all \( x \).
Distinguish between \( x^2 - 4 = 0 \) and \( x^2 - 4 \equiv (x+2)(x-2) \).
\( x^2 - 4 = 0 \) (equation): This is true only when \( x = 2 \) or \( x = -2 \). It has exactly two solutions and can be solved.
\( x^2 - 4 \equiv (x+2)(x-2) \) (identity): This is a structural fact about factorisation. It holds for every real value of \( x \) — substitute any number and both sides give the same result. There is nothing to "solve".
The identity is verified rather than solved: expand \( (x+2)(x-2) = x^2 - 2x + 2x - 4 = x^2 - 4 \). ✓
Identify the terms, coefficients and degree of \( 7a^2b - 3ab^2 + 4 \).
Terms: \( 7a^2b \), \( -3ab^2 \), \( 4 \).
Coefficients: 7 (of \( a^2b \)), \( -3 \) (of \( ab^2 \)), 4 (constant).
Degree of each term: \( a^2b \) has degree \( 2+1 = 3 \); \( ab^2 \) has degree \( 1+2 = 3 \); constant has degree 0.
Overall degree: 3 (the highest degree of any term).
Factors of \( 6x^2 \): 1, 2, 3, 6, \( x \), \( 2x \), \( 3x \), \( 6x \), \( x^2 \), \( 2x^2 \), \( 3x^2 \), \( 6x^2 \).
Key Takeaways
- An expression has no equals sign and cannot be solved; an equation can be solved; a formula is rearranged; an identity is verified for all values; an inequality gives a range of solutions.
- The identity symbol \( \equiv \) signals that the statement is true for all values of the variable — no solutions exist because it is always true.
- A term is a single component of an expression separated by \( + \) or \( - \). A coefficient is the numerical multiplier of a term.
- The degree of an expression is the highest total power of the variables in any term. Degree 1 = linear; degree 2 = quadratic; degree 3 = cubic.
- Factors of an algebraic term are all the expressions that divide it exactly.