Algebraic vocabulary

Before we start multiplying and expanding, let's give names to the building blocks. The same word can mean different things in everyday English, so being precise in mathematics saves a lot of confusion later.

Concept

Variable. A symbol (usually a letter like $x, y, a, b$) standing for an unknown or general number. In $3x + 5$, the variable is $x$.

Constant. A fixed number that does not change. In $3x + 5$, the constant is $5$.

Term. A combination of constants and variables joined only by multiplication. Examples: $3x, -7y^2, \tfrac{2}{5}ab, 4$. Terms are the "atoms" of an algebraic expression.

Coefficient. The constant in front of a term. In $-7y^2$ the coefficient is $-7$. In $\tfrac{2}{5}ab$ it is $\tfrac{2}{5}$. The coefficient of $x$ alone is $1$ (often hidden), and the coefficient of $-x$ is $-1$.

Algebraic expression. A sum or difference of terms. Examples: $3x - 7y^2 + 5$, $a + b + c$, $5x^2 - 2x + 1$.

By the number of terms, expressions are named:

• Monomial: one term. $3x, -2ab, 5$.
• Binomial: two terms. $x + 2, 3a - 4b$.
• Trinomial: three terms. $x^2 + 5x + 6, a^2 + b^2 + c^2$.
• Polynomial: any number of terms (the general word).

Like terms vs unlike terms. Two terms are like if they have the same variable parts (same letters, same powers). $5x$ and $-3x$ are like. $7x^2 y$ and $\tfrac{1}{2} x^2 y$ are like. But $5x$ and $5x^2$ are unlike , different powers.

Only like terms can be combined: $5x + (-3x) = 2x$. Unlike terms must be left alone: $5x + 7y$ does not simplify.

Degree of a term. The total power of the variables. $3x^4$ has degree $4$. $-2xy^3$ has degree $1 + 3 = 4$. A constant like $7$ has degree $0$.

Degree of a polynomial. The largest degree of its terms. $5x^3 + 2x^2 - x + 4$ has degree $3$.

Standard form. Writing a polynomial with terms in descending order of degree: $4 + 2x^2 - x + 5x^3 \to 5x^3 + 2x^2 - x + 4$.

Substitution. Replacing a variable by a number. If $p(x) = 2x^2 - 3x + 1$, then $p(4) = 2(16) - 12 + 1 = 21$. Substitution turns algebra back into arithmetic.

Worked examples

Example 1. Identify the terms, coefficients, and constant in $7x^2 - 4xy + 5$.

• Terms: $7x^2$, $-4xy$, $5$.
• Coefficients: $7$ (of $x^2$), $-4$ (of $xy$).
• Constant: $5$.

Example 2. Simplify $4x + 3y - x + 2y$.

• Like terms in $x$: $4x - x = 3x$.
• Like terms in $y$: $3y + 2y = 5y$.
• Answer: $3x + 5y$.

Example 3. Find the degree of $5a^2 b^3 - 7ab + 2$.

• Degrees of each term: $2+3 = 5$, $1+1 = 2$, $0$.
• Largest: $5$.
• Degree of polynomial $= 5$.

Example 4. Evaluate $p(x) = x^3 - 2x + 5$ at $x = -2$.

• $p(-2) = (-2)^3 - 2(-2) + 5 = -8 + 4 + 5 = 1$.

Try it yourself

1. List the terms of $-6a^2 + 3a - 7$.
2. Identify like terms among $4x^2, 5y, -2x^2, 3xy, 7$.
3. Simplify $5a + 4b - 2a - 3b$.
4. Find the coefficient of $x^2$ in $7 - 4x + 3x^2 - x^3$.
5. Classify each as monomial / binomial / trinomial: $3x$, $x + y$, $a + b + c$, $x^2 - 9$.
6. Degree of $3 x^4 y - 2x^2 y^3 + 7 x y$?
7. Evaluate $2x^2 - 5x + 3$ at $x = -1$.
8. Write $-2 + 5x^3 - 7x + x^2$ in standard form.

Activity

Term hunt. Pick any long expression you can write (or find in a worksheet). Underline each term in a different colour. Then group like terms by drawing a circle around each group. The exercise makes it visually obvious which simplifications are possible and which are not.