1.1. Introduction to Algebra
Introduction
Review of basic algebraic terms:
| Algebraic term | Description | Example |
| Algebraic expression | A mathematical phrase that contains numbers, variables (letters), and arithmetic operations (+, – , ×, ÷, etc.). |
3x – 4 5a2 – b + 3 12y3 + 7y2 – 5y + |
| Constant | A number on its own. | 2y + 5 constant: 5 |
| Coefficient | The number in front of a variable. |
-9x2 coefficient: -9 x coefficient: 1 (x = 1 · x) |
|
Term |
A term can be a constant, a variable, or the product of a number and variable. (Terms are separated by a plus or minus sign.) |
2x3 + 7x2 – 9y – 8 Terms: 2x3, 7x2, – 9y, –8 |
| Like terms | The terms that have the same variables and exponents (differ only in their coefficients). |
2x and -7x -4y2 and 9y2 0.5pq2 and pq2 |
Polynomial: an algebraic expression that contains one or more terms.
Example: 7x , 5ax – 9b , 6x2 – 5x + , 7a2 + 8b + ab – 5
There are special names for polynomials that have one, two, or three terms:
- Monomial: an algebraic expression that contains only one term.
Example: 9x , 4xy2 , 0.8mn2 , a2b
- Binomial: an algebraic expression that contains two terms.
Example: 7x + 9 , 9t2 – 2t , 0.3y +
- Trinomial: an algebraic expression that contains three terms.
Example: ax2 + bx + c , – 4qp2 + 3q + 5
Combining Terms
Like terms: terms that have the same variables and exponents (the coefficients can be different).
Examples:
| Example | Like or unlike terms |
| 7y and -9y | Like terms |
| 6a2, -32a2, and –a2 | Like terms |
| 0.3 x2y and -48x2y | Like terms |
| u2v3 and u2v3 | Like terms |
| -8y and 78x | Unlike terms |
| 6m3 and -9m2 | Unlike terms |
| -9u3w2 and -9w3u2 | Unlike terms |
Combine like terms: add or subtract their coefficients and keep the same variables and exponents.
Note: unlike terms cannot be combined.
Example 1.1.1
| 1) 3a + 7b – 9a + 15b = (3a – 9a) + (7b + 15b) | Regroup like terms. |
| = -6a + 22b | Combine like terms. |
| 2) 2y2 – 4x + 3x – 5y2 = (2y2 – 5y2) + (-4x + 3x) | Regroup like terms. |
| = -3y2 – 1x | Combine like terms. |
| = -3y2 – x |
| 3) 8xy2 – x2y + 4x2y – 6xy2 | |
| = 8xy2 – x2y + 4x2y – 6xy2 | Or underline like terms without regrouping. |
| = 2xy2 + 3x2y | Combine like terms. |
| 4) 2(2m + 3n) + 3(m – 4n) = 4m + 6n + 3m – 12n | Distributive property. |
| = 7m – 6n | Combine like terms. |
Removing Parentheses
If the sign preceding the parentheses is positive (+), do not change the sign of terms inside the parentheses, just remove the parentheses.
If the sign preceding the parentheses is negative (-), remove the parentheses and the negative sign (in front of parentheses), and change the sign of each term inside the parentheses.
Remove parentheses:
| Algebraic expression | Remove parentheses | Example |
| (ax + b) | ax + b | (5x + 2) = 5x + 2 |
| (ax – b) | ax – b | (9y – 4) = 9y – 4 |
| – (ax + b) | -ax – b | – (x + 7) = – x – 7 |
| – (ax – b) | -ax + b | – (0.5b – 2.4) = -0.5b + 2.4 |
Example 1.1.2
| 1) 9x2 + 7 – (2x2 – 2) = 9x2 + 7 – 2x2