Algebra 1 Section 2.3

The Distributive Property For any real numbers a, b, and c, a(b + c) = ab + ac The distributive property is often used to simplify algebraic expressions.

Example 1 Distribute 4(a + 3b). 4(a + 3b) 4(a) + 4(3b) 4a + 12b

Example 2 Distribute -2(3x2 – 4x + 5). -2(3x2) – (-2)(4x) + (-2)(5)

Definition A term is a constant, variable raised to a power, or the product or quotient of a constant and one of more such variables.

Definition Like terms are terms that have the same variable (or variables) with corresponding variables having the same exponents.

More About Terms Terms of an expression are separated by addition or subtraction. 3x – 4y + 5x This expression has three terms.

Example 3 Simplify 3x – 4y + 5x. 3x + 5x – 4y (3 + 5)x – 4y 8x – 4y

Example 4 Simplify 7x2 + 8x – 4x2 + 10x. 7x2 – 4x2 + 8x + 10x

Definition The numerical coefficient (often referred to as the coefficient) is the numerical factor (constant) accompanying the variable(s) in a term.

More About Coefficients 3x2 – 7x2 + x2 The coefficient of... 3x2 is 3. -7x2 is -7. x2 is 1.

Combining Like Terms Identify like terms. Combine like terms by adding their coefficients. 3x2 – 7x2 + x2 [3 + (-7) + 1]x2 -3x2

Example 5 Simplify 8x3 – x2 + x3 + 4x3 – x2 + 6x3.

Example 6 Simplify 4x2y + 5xy – 10x2y + 22xy – 8x + y.

Example 7 Simplify 4x2yz + 2x2y + xy2 – 3xyz. The expression contains no like terms. It cannot be simplified further.

Example 8 Simplify (2x + 5) + (3x – 9) – (x – 6). 2x + 5 + 3x – 9 – x + 6 4x + 2

Example 9 Simplify 3k(k + 2) – k(k – 1). 3k2 + 6k – k2 + k 2k2 + 7k

Example 10 7x2 – [3x + 4y(2x – 8y) – 4x] + 5xy 7x2 – [3x + 8xy – 32y2 – 4x] + 5xy 7x2 – [-x + 8xy – 32y2] + 5xy 7x2 + x – 8xy + 32y2 + 5xy 7x2 – 3xy + x + 32y2

Homework: pp. 60-61