Building expressions
A variable is a letter that can take different values, such as x, y or n. A constant has a fixed value, such as 5 or −3. An algebraic expression combines variables and constants using +, −, × and ÷.
In words | Expression |
|---|---|
5 more than a number x | x + 5 |
3 times a number y, minus 4 | 3y − 4 |
half of the product of p and q | pq/2 |
the square of m, plus 1 | m² + 1 |
Terms, factors and coefficients
The parts of an expression joined by + or − are its terms. The expression 4x² − 3xy + 7 has three terms: 4x², −3xy and 7.
Each term is a product of factors. The term −3xy has factors −3, x and y.
The coefficient is the numerical factor of a term: the coefficient of x² in 4x² is 4, and of xy in −3xy is −3. When no number is written, the coefficient is 1: the coefficient of y in y is 1, and in −y it is −1.
Like and unlike terms
Like terms have exactly the same variables raised to the same powers. Only the coefficients may differ.
5x and −2x are like terms.
3ab and 7ba are like terms (ab = ba).
4x and 4x² are unlike, because the powers differ.
2xy and 2x are unlike, because the variables differ.
Types of expressions
Name | Number of terms | Example |
|---|---|---|
Monomial | 1 | 7xy |
Binomial | 2 | a + 4 |
Trinomial | 3 | x² + 2x − 5 |
Polynomial | one or more | 3p³ − p² + 2p + 1 |
Adding and subtracting
Only like terms can be combined. Add or subtract their coefficients and keep the variable part.
7x + 3x = 10x and 5ab − 8ab = −3ab, but 3x + 4y stays as it is.
Example. Add 3x² − 2x + 5 and x² + 6x − 9.
Group like terms: (3x² + x²) + (−2x + 6x) + (5 − 9) = 4x² + 4x − 4.
Example. Subtract 2a − 3b + 4 from 5a + b − 1.
Change the sign of every term being subtracted, then add:
(5a + b − 1) − (2a − 3b + 4) = 5a + b − 1 − 2a + 3b − 4 = 3a + 4b − 5.
Finding the value of an expression
Replace each variable with its given value, then calculate, following the order of operations.
Example. Find the value of 2x² − 3x + 1 when x = −2.
2(−2)² − 3(−2) + 1 = 2 × 4 + 6 + 1 = 15.
Put negative values in brackets so the signs stay correct: (−2)² = 4, not −4.
Using expressions for patterns
Expressions describe rules. A row of squares made with matchsticks needs 4 sticks for one square, 7 for two, and 10 for three. Each new square adds 3 sticks, so n squares need 3n + 1 sticks. For 20 squares, that is 3 × 20 + 1 = 61 sticks.
Common mistakes
Combining unlike terms: 3x + 2y is not 5xy, and x + x² is not 2x³.
Forgetting to change every sign when subtracting a bracket.
Writing the coefficient of x as 0. In x it is 1.
Summary
Terms are separated by + and −; a coefficient is the numerical factor of a term.
Like terms have identical variable parts and can be combined.
To subtract an expression, change the sign of each of its terms and add.