Algebraic Expressions: Grade 9 Notes

1. Definition

An algebraic expression is a combination of variables, numbers, and operations (addition, subtraction, multiplication, and division). It does not have an equality sign.

2. Components of Algebraic Expressions

Variables: Symbols (like x, y, z) that represent unknown values.
Constants: Fixed values (like 2, -5, 10).
Coefficients: Numbers multiplying the variables (in 3x, 3 is the coefficient).
Terms: Parts of an expression separated by + or - (e.g., in 4x + 5, there are two terms: 4x and 5).
Operators: Symbols representing operations (e.g., +, -, *, /).

3. Types of Algebraic Expressions

Monomial: An expression with one term (e.g., 3x, -7y^2).
Binomial: An expression with two terms (e.g., 4x + 5, 3a - 2b).
Trinomial: An expression with three terms (e.g., x^2 + 3x + 2).

4. Like Terms

Terms that have the same variables raised to the same powers (e.g., 5x and 3x are like terms; 4x^2 and 7x^2 are like terms).

5. Combining Like Terms

To simplify an expression, add or subtract the coefficients of like terms.

Example: 3x + 4x = 7x; 5a^2 - 2a^2 = 3a^2.

6. Simplifying Algebraic Expressions

Combine like terms.
Use the distributive property: a(b + c) = ab + ac.

Example: 2(x + 3) = 2x + 6.

7. Expanding Algebraic Expressions

Use the distributive property to remove parentheses.

Example: (x + 2)(x - 3) = x^2 - 3x + 2x - 6 = x^2 - x - 6.

8. Factoring Algebraic Expressions

The process of writing an expression as a product of its factors. Common methods include:

Factoring out the Greatest Common Factor (GCF): e.g., 6x + 9 = 3(2x + 3).
Factoring Trinomials: e.g., x^2 + 5x + 6 = (x + 2)(x + 3).
Difference of Squares: e.g., x^2 - 9 = (x + 3)(x - 3).

9. Evaluating Algebraic Expressions

Substitute the given values for variables and perform the operations.

Example: For the expression 3x + 4, if x = 2, then 3(2) + 4 = 6 + 4 = 10.

10. Working with Exponents

Power Rules:

a^m * a^n = a^(m+n)
(a^m)^n = a^(m*n)

Example: x^3 * x^2 = x^(3+2) = x^5; (2x)^3 = 2^3 * x^3 = 8x^3.

11. Special Products

Square of a Binomial: (a + b)^2 = a^2 + 2ab + b^2; (a - b)^2 = a^2 - 2ab + b^2.
Product of Sum and Difference: (a + b)(a - b) = a^2 - b^2.