Algebraic Expressions

Principles of Algebra · 3 min read

Algebraic Expressions

Understanding Variables and Constants

A variable is a symbol, typically a letter, that represents an unknown or changeable quantity. Common variables include x, y, z, a, b, and n. A constant is a fixed value that does not change. In the expression 3x + 7, the number 3 is the coefficient of x, x is the variable, and 7 is the constant term.

An algebraic expression is a combination of variables, constants, and operations (addition, subtraction, multiplication, division, exponentiation). Unlike equations, expressions do not contain an equals sign. Examples include 2x + 5, 4a² − 3a + 1, and (x + y)/2.

Terms and Like Terms

A term is a single number, variable, or the product of numbers and variables. In the expression 5x² + 3x − 2, there are three terms: 5x², 3x, and −2. Terms are separated by addition or subtraction signs.

Like terms have identical variable parts with identical exponents. For example, 4x² and −7x² are like terms because both contain x². However, 4x² and 4x are not like terms because the exponents differ. Constants are like terms with each other.

To combine like terms, add or subtract their coefficients while keeping the variable part unchanged. For example: 5x + 3x = 8x, and 7y² − 2y² = 5y².

Evaluating Expressions

To evaluate an algebraic expression, substitute the given values for each variable and perform the indicated operations following the order of operations (PEMDAS: Parentheses, Exponents, Multiplication/Division left to right, Addition/Subtraction left to right).

Worked Example

Evaluate 3x² − 2x + 4 when x = 5. Substitute: 3(5)² − 2(5) + 4 = 3(25) − 10 + 4 = 75 − 10 + 4 = 69.

Translating Verbal Phrases

Translating words into algebraic expressions is a fundamental skill. Key phrases include:

Addition

sum, plus, increased by, more than, total

Subtraction

difference, minus, decreased by, less than, fewer than

Multiplication

product, times, of, twice, double

Division

quotient, divided by, ratio, per

Worked Example

"Five less than three times a number" translates to 3x − 5. Note that "less than" reverses the order of subtraction.

Properties of Real Numbers

The following properties govern operations with real numbers and algebraic expressions:

Commutative Property

Order does not affect the result for addition and multiplication. a + b = b + a and ab = ba.

Associative Property

Grouping does not affect the result for addition and multiplication. (a + b) + c = a + (b + c) and (ab)c = a(bc).

Distributive Property

Multiplication distributes over addition and subtraction. a(b + c) = ab + ac and a(b − c) = ab − ac.

Identity Property

Adding zero or multiplying by one leaves a number unchanged. a + 0 = a and a × 1 = a.

Inverse Property

A number plus its opposite equals zero; a number times its reciprocal equals one. a + (−a) = 0 and a × (1/a) = 1 (where a ≠ 0).

Simplifying Expressions

To simplify an algebraic expression, apply the distributive property to remove parentheses, then combine like terms.

Worked Example

Simplify 4(2x − 3) + 5x − 2. Apply distributive property: 8x − 12 + 5x − 2. Combine like terms: 13x − 14.

When an expression contains nested parentheses, work from the innermost grouping outward.

Worked Example

Simplify 2[3(x + 4) − 5]. Inner parentheses first: 2[3x + 12 − 5] = 2[3x + 7]. Apply distribution: 6x + 14.

Exponent Rules for Expressions

Integer exponents follow specific rules essential for simplifying expressions:

Product Rule

When multiplying powers with the same base, add exponents. x^m × x^n = x^(m+n).

Quotient Rule

When dividing powers with the same base, subtract exponents. x^m ÷ x^n = x^(m−n).

Power Rule

When raising a power to a power, multiply exponents. (x^m)^n = x^(mn).

Power of a Product

(xy)^n = x^n × y^n.

Power of a Quotient

(x/y)^n = x^n/y^n.

Zero Exponent

x^0 = 1 (where x ≠ 0).

Negative Exponent

x^(−n) = 1/x^n.

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← Exam Overview

Algebraic Expressions

Understanding Variables and Constants

Terms and Like Terms

To combine like terms, add or subtract their coefficients while keeping the variable part unchanged. For example: 5x + 3x = 8x, and 7y² − 2y² = 5y².

Evaluating Expressions

Worked Example

Evaluate 3x² − 2x + 4 when x = 5. Substitute: 3(5)² − 2(5) + 4 = 3(25) − 10 + 4 = 75 − 10 + 4 = 69.

Translating Verbal Phrases

Translating words into algebraic expressions is a fundamental skill. Key phrases include:

Addition

sum, plus, increased by, more than, total

Subtraction

difference, minus, decreased by, less than, fewer than

Multiplication

product, times, of, twice, double

Division

quotient, divided by, ratio, per

Worked Example

"Five less than three times a number" translates to 3x − 5. Note that "less than" reverses the order of subtraction.

Properties of Real Numbers

The following properties govern operations with real numbers and algebraic expressions:

Commutative Property

Order does not affect the result for addition and multiplication. a + b = b + a and ab = ba.

Associative Property

Grouping does not affect the result for addition and multiplication. (a + b) + c = a + (b + c) and (ab)c = a(bc).

Distributive Property

Multiplication distributes over addition and subtraction. a(b + c) = ab + ac and a(b − c) = ab − ac.

Identity Property

Adding zero or multiplying by one leaves a number unchanged. a + 0 = a and a × 1 = a.

Inverse Property

A number plus its opposite equals zero; a number times its reciprocal equals one. a + (−a) = 0 and a × (1/a) = 1 (where a ≠ 0).

Simplifying Expressions

To simplify an algebraic expression, apply the distributive property to remove parentheses, then combine like terms.

Worked Example

Simplify 4(2x − 3) + 5x − 2. Apply distributive property: 8x − 12 + 5x − 2. Combine like terms: 13x − 14.

When an expression contains nested parentheses, work from the innermost grouping outward.

Worked Example

Simplify 2[3(x + 4) − 5]. Inner parentheses first: 2[3x + 12 − 5] = 2[3x + 7]. Apply distribution: 6x + 14.

Exponent Rules for Expressions

Integer exponents follow specific rules essential for simplifying expressions:

Product Rule

When multiplying powers with the same base, add exponents. x^m × x^n = x^(m+n).

Quotient Rule

When dividing powers with the same base, subtract exponents. x^m ÷ x^n = x^(m−n).

Power Rule

When raising a power to a power, multiply exponents. (x^m)^n = x^(mn).

Power of a Product

(xy)^n = x^n × y^n.

Power of a Quotient

(x/y)^n = x^n/y^n.

Zero Exponent

x^0 = 1 (where x ≠ 0).

Negative Exponent

x^(−n) = 1/x^n.

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Quiz me Next chapter →

Praxis Algebra I (5162)