Algebraic Expressions: Foundations, Evaluation, and Simplification

Algebraic Expressions

Definition and Components

Algebraic expressions are fundamental objects in algebra, consisting of numbers, variables, and mathematical operations. Understanding their structure is essential for manipulating and evaluating expressions in College Algebra.

• Algebraic Expression: A combination of numbers and variables connected by mathematical operations (addition, subtraction, multiplication, division, exponents).

• Variable: A letter (such as x, y, a) representing an unknown or changeable value.

• Coefficient: The numerical factor multiplying a variable. For example, in $3x$, 3 is the coefficient.

• Constant: A number without variables; its value does not change. For example, in $2x + 5$, 5 is the constant.

Example: In the expression $2x + 5$:

• Numbers? Yes (2, 5)

• Operations? Yes (addition, multiplication)

• Variables? Yes (x)

• Coefficient: 2

• Constant: 5

When an equality symbol (=) is placed between two expressions, it forms an equation.

Evaluating Algebraic Expressions

To evaluate an algebraic expression, substitute the given values for the variables and perform the operations according to the order of operations (PEMDAS).

• Order of Operations:

  1. Parentheses

  2. Exponents

  3. Multiplication/Division

  4. Addition/Subtraction

• Example: Evaluate $2x + 5$ when $x = 3$:

  • Substitute: $2 \times 3 + 5$

  • Calculate: $6 + 5 = 11$

• Example: Evaluate $4\sqrt{x} + 8$ when $x = 9$:

  • Substitute: $4 \sqrt{9} + 8$

  • Calculate: $4 \times 3 + 8 = 12 + 8 = 20$

Exponents in Expressions

Exponents are used to represent repeated multiplication of a base number. They are a key part of algebraic expressions and must be handled according to the order of operations.

• Base: The number or variable being multiplied.

• Exponent (Power): Indicates how many times the base is multiplied by itself.

• General Form: $a \cdot a \cdot \ldots \cdot a$ ($n$ times) $= a^n$

• Example: $4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 = 4^5$

• Example: $x^3$ means $x \cdot x \cdot x$

Simplifying Algebraic Expressions

Simplifying expressions involves reducing them to their simplest form by combining like terms and applying distributive properties. This process makes expressions easier to work with and solve.

• Term: A part of an expression separated by plus (+) or minus (−) signs.

• Like Terms: Terms with the same variable(s) raised to the same power(s).

• Steps to Simplify:

  1. Distribute constants/variables through parentheses (if any).

  2. Group like terms by writing them next to each other.

  3. Combine like terms by adding or subtracting their coefficients.

• Example: Simplify $2x + 3 + 4(x + 2)$:

  • Distribute: $2x + 3 + 4x + 8$

  • Group like terms: $2x + 4x + 3 + 8$

  • Combine: $6x + 11$

• Example: Simplify $-3(5 - x) + 10 - 7x$:

  • Distribute: $-15 + 3x + 10 - 7x$

  • Group like terms: $3x - 7x - 15 + 10$

  • Combine: $-4x - 5$

Practice Problems

• Evaluate $2y - x(3 + y)$ when $x = 2$, $y = 5$:

  • Substitute: $2 \times 5 - 2(3 + 5)$

  • Calculate: $10 - 2 \times 8 = 10 - 16 = -6$

• Simplify $3x + 14y - 7(-x + 2y)$:

  • Distribute: $3x + 14y + 7x - 14y$

  • Group like terms: $3x + 7x + 14y - 14y$

  • Combine: $10x$

Terms in Expressions

Terms can be classified as follows:

Type	Example	Description
Variable only	x	Contains only a variable
Number and variable	3y	Contains a coefficient and a variable
Number only (constant)	5	Contains only a number

Additional info: The notes emphasize the importance of identifying and combining like terms, as well as following the order of operations for accurate evaluation and simplification. These skills are foundational for solving equations and more advanced algebraic manipulations.