Chapter 11.1 — Introduction to Algebraic Expressions
Algebraic expressions — addition, subtraction and multiplication of monomials. This is Lesson 1 of 6 in Chapter 11: Algebraic Expressions.
Naming the Building Blocks First
Before combining or simplifying anything, algebra needs names for its own ingredients. A variable is a letter standing in for a number that isn't fixed — x, y, a, b. A constant is the opposite: a specific, unchanging number or symbol, like 3, −9, 8.5, or π. A term is a number, a variable, or a product of numbers and variables multiplied together — 5, t, 2x, 3.5pq, and (3/2)lm are all single terms. An expression (or algebraic expression) is what you get by adding or subtracting terms: x + 5y − 1, or 2a² + 3a + 5.
Numerical Versus Algebraic
Not everything built from numbers and operations counts as algebraic. 3 + 5 − 0.5 + √3 uses no variable at all, so it's a numerical expression, not an algebraic one — the presence of at least one variable is what earns an expression that label.
Counting Terms Tells You the Name
Once an expression is broken into its added or subtracted terms, the count of those terms gives it a family name:
| Terms | Name | Example |
|---|---|---|
| Exactly 1 | Monomial | 5xy² |
| Exactly 2 | Binomial | 5xy³ − 9x |
| Exactly 3 | Trinomial | 3xy + 4y − 8 |
| More than 3 | Multinomial | 9x² + 2x + pq + q |
This naming only counts terms separated by + or − at the outermost level — the fact that a single term like 3.5pq itself contains a product of three symbols doesn't turn it into three terms; it's still one monomial.
Counting Terms Correctly, in Practice
The naming rule above is easy to state but worth practising directly, since it's easy to miscount when several variables are mixed into one term:
| Expression | Terms | Name |
|---|---|---|
| 5xy² | 1 | Monomial |
| 5xy³ − 9x | 2 | Binomial |
| 3xy + 4y − 8 | 3 | Trinomial |
| 9x² + 2x + pq + q | 4 | Multinomial |
The first row is the one most often miscounted: 5xy² looks "busy" because it has two different variables in it, but there's no + or − anywhere inside it — it's one product multiplied together, and therefore exactly one term, not two.
Degree: How Much "Power" Sits in a Term
The degree of a monomial is the sum of all the exponents on its variables:
| Monomial | Degree |
|---|---|
| 5xy² | 1 + 2 = 3 |
| −3l³m²n | 3 + 2 + 1 = 6 |
| pq | 1 + 1 = 2 |
| 3a | 1 |
| 1.5 | 0 (a constant has no variable to raise to a power) |
For a full expression with several terms, the degree of the expression is simply the highest degree among all its individual terms:
| Expression | Degree |
|---|---|
| 3x² − 5x + 6 | 2 |
| 5pqr − q² | 3 |
| 3x² − x³ + x + 1 | 3 |
| 7 + a | 1 |
| 3x³y + 5xy − 4 | 4 |
Notice the last row: 3x³y has degree 3 + 1 = 4, which is higher than 5xy's degree of 2 — the expression's overall degree comes from whichever single term contributes the most exponents added together, not from the term that looks longest on the page.
Same Letters, Same Powers: What Makes Terms "Like"
Terms are like terms when they share the same variables raised to the same exponents — 3x², −5x², 21x², (1/6)x², and 0.9x² are all like terms, even though their numerical coefficients are completely different. Anything that fails that variable-and-exponent match is an unlike term: 3x², −5x³, and 21p² are unlike terms, since the exponents on x differ between the first two, and the variable itself differs for the third. Only like terms can ever be combined into a single term; unlike terms can sit side by side in a simplified expression, but they can never be merged into one.
These Names Aren't New — Just Newly Labelled
Every formula from earlier chapters already used this vocabulary without naming it. The area of a rectangle, l × b, is a monomial in two variables — one single term, degree 1 + 1 = 2. The trapezium area formula, ½ × (a + b) × h, expands into a two-term (binomial) expression once the bracket is multiplied through: ½ah + ½bh. Even a plain number like the ½ in that formula is a constant sitting alongside the variables a, b, and h. Naming these parts explicitly now — variable, constant, term, degree — is what makes it possible to talk precisely about how expressions combine, rather than relying on "that formula thing with the letters," which is exactly what the rest of this chapter needs in order to multiply and simplify expressions systematically instead of case by case.
Adding and Subtracting by Matching Like Terms
Combining expressions is just combining like terms, one group at a time. Adding 5a² − 3ab + 7b² and 8a² + 2ab + 3b²:
(5a² − 3ab + 7b²) + (8a² + 2ab + 3b²)
= (5a² + 8a²) + (−3ab + 2ab) + (7b² + 3b²)
= 13a² − ab + 10b²Subtraction works the same way, but every sign in the expression being subtracted flips first. Subtracting 3x² + y² − xy from x² + 2xy:
(x² + 2xy) − (3x² + y² − xy)
= x² + 2xy − 3x² − y² + xy
= −2x² + 3xy − y²That sign flip is the single most common place a subtraction goes wrong — dropping it turns a subtraction into an addition halfway through the problem, silently changing the answer, since −(3x²+y²−xy) is not the same expression as −3x²+y²−xy no matter how similar the two look side by side.
Multiplying Two Monomials
Multiplying monomials combines their numbers and combines their matching variables' exponents separately:
5p × 7q = (5 × 7) × (p × q) = 35pq
−2a²b × 4a × 8b² = (−2 × 4 × 8) × (a² × a) × (b × b²) = −64a³b³In the second example, a² and a combine into a³ (exponents 2 + 1), and b and b² combine into b³ (exponents 1 + 2) — the numerical part and each variable's exponents are handled as entirely separate mini-calculations that only come back together at the very last step. This is also a convenient moment to notice how degree behaves under multiplication: 5p has degree 1 and 7q has degree 1, and their product 35pq has degree 2 — the degrees simply add. The same check works on the second example: −2a²b has degree 3, 4a has degree 1, and 8b² has degree 2, adding up to 6, exactly matching the degree of the final product −64a³b³ (3 + 3 = 6).
Where Multiplication Goes From Here
Multiplying a monomial by another monomial is the simplest of several multiplication patterns this chapter builds toward — Exercise 11.1 practices exactly that skill on a wider range of examples, including three-and-more-factor products and real box-volume problems. From there, the chapter works up to multiplying a monomial by a whole polynomial, then a binomial by another binomial or trinomial, before arriving at a set of algebraic identities that turn some of the most frequent multiplications into instant shortcuts, and finally showing why those same identities are also true geometrically, not just algebraically.