Algebraic Expressions: Structure, Evaluation, and Simplification

Algebraic Expressions

Definition and Components

An algebraic expression is a combination of numbers and variables connected by mathematical operations such as addition, subtraction, multiplication, and division.

• Variable: A letter that represents any number; its value can vary.

• Coefficient: A number multiplying a variable; its value can change if the variable changes.

• Constant: A number without variables; its value does not change.

Algebraic expressions differ from numerical expressions, which contain only numbers and operations.

Numerical Expressions	Algebraic Expressions
2(3) + 5	2x + 5
Numbers, operations	Numbers, operations, variables

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

• 2 is the coefficient

• x is the variable

• 5 is the constant

Identifying Coefficients and Constants

To analyze algebraic expressions, identify the coefficients (numbers multiplying variables) and constants (numbers without variables).

Expression	Coefficient	Constant
$4x^2 + 8$	4	8
$3(4 + 5x)$	15 (after distributing: $3 \times 5$)	12 (after distributing: $3 \times 4$)
$2 - 3xy$	-3	2
$9x + 18$	9	18

Evaluating Algebraic Expressions

Substituting Values

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

• Parentheses

• Exponents

• Multiply/Divide

• Add/Subtract

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

• Substitute: $2(3) + 5$

• Calculate: $6 + 5 = 11$

Example: Evaluate $2y - x(3 + y)$ when $x = 4$, $y = -5$:

• Substitute: $2(-5) - 4(3 + (-5))$

• Calculate: $-10 - 4(-2) = -10 + 8 = -2$

Exponents in Expressions

Definition and Notation

An exponent (or power) represents repeated multiplication of the same number or variable.

• Base: The number or variable being multiplied.

• Exponent: The number of times the base is multiplied by itself.

General Form:

• $a^n = a \times a \times \cdots \times a$ ($n$ times)

Example: $4^5 = 4 \times 4 \times 4 \times 4 \times 4$

Example: $x^3 = x \times x \times x$

Evaluating Expressions with Exponents

Follow the order of operations, calculating exponents before multiplication or addition.

• Example: Evaluate $-3x^4$ when $x = 2$:

• Calculate: $-3(2^4) = -3(16) = -48$

Simplifying Algebraic Expressions

Terms and Like Terms

Algebraic expressions are made up of terms, which are separated by plus or minus signs. Like terms have the same variable(s) raised to the same power.

• Term: A part of an expression separated by $+$ or $-$

• Like Terms: Terms with the same variable(s) and exponent(s)

Example: In $5 - x + 3y + y$:

• 5 is a constant term

• -x is a variable term

• 3y and y are like terms

Steps to Simplify Algebraic Expressions

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/subtracting their coefficients

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

• Distribute: $2x + 3 + 4x + 8$

• Group: $(2x + 4x) + (3 + 8)$

• Combine: $6x + 11$

Practice Problems

• Simplify $-3(5 - x) + 10 - 7x$

• Simplify $-13 + 4x + 9(-x)$

• Simplify $3x + 14y - 7(-x + 2y)$

Follow the steps above to solve each.

Order of Operations (PEMDAS)

Summary Table

Order	Operation
1	Parentheses
2	Exponents
3	Multiply/Divide
4	Add/Subtract

Always follow this order when evaluating or simplifying expressions.

Additional info: Some context and explanations have been expanded for clarity and completeness, including the explicit steps for identifying coefficients, constants, and simplifying expressions.