Summary: Algebraic Expressions

Table of Contents
1. Introduction
2. Types of Algebraic Expressions
3. Parts of an Algebraic Expression
4. Evaluating Algebraic Expressions
5. Simplifying Algebraic Expressions (Combining Like Terms)
6. Addition and Subtraction of Algebraic Expressions
7. Degree of a Term and of a Polynomial
8. Classification by Degree and Number of Terms
9. Tips for Working with Algebraic Expressions
10. Summary
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Introduction

• An algebraic expression is a mathematical phrase formed from variables and constants using arithmetic operations such as addition, subtraction, multiplication and division.
• Variables are symbols used to represent numbers; they are usually small letters such as x, y, z, a, b, c and can take different numerical values.
• The separate parts of an expression which are first formed and then added (or subtracted) are called terms. Terms are joined by + or - signs to form an expression.
• A coefficient is the numerical factor of a term. In 7x, the coefficient of x is 7. Sometimes any factor in a term is called the coefficient of the remaining part of the term.
• Terms having the same algebraic factors are called like terms.
• Terms having different algebraic factors are called unlike terms.

Types of Algebraic Expressions

• Monomial: An expression with one term. Example: 5x, -3a², 7
• Binomial: An expression with two unlike terms. Example: x + 2, 3x² - 5x
• Trinomial: An expression with three unlike terms. Example: x² + 5x + 6
• Polynomial: An expression with one or more terms. A monomial, a binomial and a trinomial are all polynomials.

Parts of an Algebraic Expression

• Constant: A term with no variable. Example: 4, -7
• Variable: A symbol representing an unknown or changing number. Example: x, y
• Coefficient: The number multiplying the variable part of a term. Example: in 6xy, the coefficient is 6
• Term: A product of numbers and variables. Terms are separated by + or -. Example: in 3x + 4y - 5, the terms are 3x, 4y and -5
• Like terms: Terms with the same variable factors. Example: 4x and -7x are like terms. 3ab and -5ab are like terms.
• Unlike terms: Terms with different variable factors. Example: 2x and 3y are unlike terms. 5x² and 2x are unlike terms.

Evaluating Algebraic Expressions

To evaluate an algebraic expression, substitute the given numerical values for the variables and simplify.

Example 1: Evaluate 3x + 5 when x = 2.

Substitute x = 2 in the expression 3x + 5.

3×2 + 5

6 + 5

11

Example 2: Evaluate 2a² - 3b when a = 2 and b = 1.

Substitute a = 2 and b = 1 in the expression 2a² - 3b.

2×(2)² - 3×1

2×4 - 3

8 - 3

5

Simplifying Algebraic Expressions (Combining Like Terms)

Simplifying means writing an expression in its simplest form by combining like terms and performing arithmetic on coefficients.

Example 3: Simplify 5x + 3y - 2x + 7y.

Group like terms together.

(5x - 2x) + (3y + 7y)

3x + 10y

Example 4: Simplify 4a - 2a + 6 - 3.

Group like terms together.

(4a - 2a) + (6 - 3)

2a + 3

Addition and Subtraction of Algebraic Expressions

• The sum of two like terms is a like term whose coefficient is the sum of the coefficients of the two like terms.
• When we add two algebraic expressions, we add their like terms and write unlike terms as they are.
• The difference of two like terms is a like term whose coefficient is the difference of the coefficients of the two like terms.
• When we subtract two algebraic expressions, we subtract their like terms and write unlike terms as they are (with appropriate signs).

Example 5: Add (2x² + 3x - 5) and (x² - x + 8).

Write the expressions one after the other and combine like terms.

(2x² + 3x - 5) + (x² - x + 8)

2x² + x² + 3x - x - 5 + 8

3x² + 2x + 3

Example 6: Subtract (x² + 4x - 3) from (3x² - 2x + 5).

Compute (3x² - 2x + 5) - (x² + 4x - 3).

3x² - 2x + 5 - x² - 4x + 3

(3x² - x²) + (-2x - 4x) + (5 + 3)

2x² - 6x + 8

Degree of a Term and of a Polynomial

• Degree of a term (single-variable): The power of the variable in that term. Example: degree of 7x³ is 3.
• Degree of a polynomial: The highest degree among its terms. Example: for 4x³ + 2x² - x + 7, the degree is 3.
• For terms with more than one variable, the degree of the term is the sum of powers of all variables in that term. Example: degree of 3x²y is 3 (2 + 1).

Classification by Degree and Number of Terms

• A polynomial of degree 0 is a constant (e.g., 5).
• A polynomial of degree 1 is linear (e.g., 3x + 2).
• A polynomial of degree 2 is quadratic (e.g., x² + 5x + 6).
• Classification by number of terms: monomial (1), binomial (2), trinomial (3), polynomial (one or more).

Tips for Working with Algebraic Expressions

• Always identify and group like terms before combining them.
• Carefully apply signs when subtracting expressions-distribute the negative sign to every term of the subtracted expression.
• When evaluating, substitute the given numbers first, then follow the order of operations (brackets, indices/powers, multiplication/division, addition/subtraction).
• Keep expressions in a standard order (highest power first) to read degrees quickly.

Summary

An algebraic expression is built from variables, constants and arithmetic operations. Know how to identify terms, coefficients, like and unlike terms, and the degree of terms and polynomials. Simplifying, evaluating, adding and subtracting expressions all depend on correctly grouping and operating on like terms. Practice with substitution and combination of terms to gain confidence.