Order of Operations and Simplifying Expressions: College Algebra Study Notes

Order of Operations and Simplifying Expressions

Introduction

This section introduces fundamental algebraic concepts, focusing on evaluating and simplifying algebraic expressions, understanding variables, and applying mathematical models. Mastery of these skills is essential for success in College Algebra and forms the basis for more advanced topics.

Algebraic Expressions

Variables and Expressions

• Variable: A letter used to represent various numbers in mathematical expressions.

• Algebraic Expression: A combination of variables and numbers using operations such as addition, subtraction, multiplication, division, powers, or roots.

• Exponential Expression: An expression of the form $b^n$, where $b$ is the base and $n$ is the exponent.

Evaluating an Algebraic Expression means finding its value for a given value of the variable.

Example: Evaluating an Algebraic Expression

• Evaluate $8 + 6(x - 3)^2$ for $x = 13$.

• Solution: Substitute $x = 13$ into the expression:

$8 + 6(13 - 3)^2 = 8 + 6(10)^2 = 8 + 6(100) = 8 + 600 = 608$

Checkpoint 1

• Evaluate $2 + 3(x - 1)^2 - 4$ for $x = 3$.

• Solution:

$2 + 3(3 - 1)^2 - 4 = 2 + 3(2)^2 - 4 = 2 + 3(4) - 4 = 2 + 12 - 4 = 10$

Formulas and Mathematical Models

Definitions and Applications

• Equation: A statement formed when an equal sign is placed between two algebraic expressions.

• Formula: An equation that uses variables to express a relationship between two or more quantities.

• Mathematical Modeling: The process of finding formulas to describe real-world phenomena. The resulting formulas, along with the meaning assigned to the variables, are called mathematical models.

Example: Using a Mathematical Model

• The formula $T = 4x^2 + 341x + 3194$ models the average cost of tuition and fees, $T$, for public U.S. colleges for the school year ending $x$ years after 2000.

• To project the average cost for the year 2015, substitute $x = 15$:

$T = 4(15)^2 + 341(15) + 3194 = 4(225) + 5115 + 3194 = 900 + 5115 + 3194 = 9209$

Additional info: This demonstrates how mathematical models are used to make predictions based on real-world data.

Checkpoint 2

• Use the formula above to find the average cost for the year 2010 ($x = 10$):

$T = 4(10)^2 + 341(10) + 3194 = 4(100) + 3410 + 3194 = 400 + 3410 + 3194 = 7004$

Simplifying Algebraic Expressions

Combining Like Terms

• Like Terms: Terms that have exactly the same variable factors (including the same exponents).

• Simplified Expression: An algebraic expression is simplified when all parentheses have been removed and like terms have been combined.

Example: Simplifying an Algebraic Expression

• Simplify $7(4x^2 + 3x) + 2(5x^2 + x)$

• Solution:

$7(4x^2 + 3x) + 2(5x^2 + x) = 28x^2 + 21x + 10x^2 + 2x = (28x^2 + 10x^2) + (21x + 2x) = 38x^2 + 23x$

Checkpoint 7

• Simplify $6(2x^2 + 4x) + 10(4x^2 + 3x)$

• Solution:

$6(2x^2 + 4x) + 10(4x^2 + 3x) = 12x^2 + 24x + 40x^2 + 30x = (12x^2 + 40x^2) + (24x + 30x) = 52x^2 + 54x$

Summary Table: Checkpoint Answers

Checkpoint	Answer
CP 1	10
CP 2	T = 7004
CP 7	52x^2 + 54x