1. Variables, Constants, and Algebraic Symbols

Variables

A variable is a symbol (usually a letter like x, y, or z) that can represent different numerical values. Variables allow us to generalize mathematical statements and express formulas for a wide range of scenarios.

Constants

A constant is a fixed value that does not change. Examples include numbers like 2, -5, or ½.

Algebraic Symbols

Letters, numbers, and operation signs (+, −, ×, ÷) together make up the language of algebra. For instance, in 3x + 4, “3” and “4” are constants, “x” is a variable, and “+” is an operation.

2. Terms, Factors, and Coefficients

Term

A term is a single part of an expression that may include a number, a variable, or a combination multiplied together. For example, in 7x², “7x²” is a term.

Factor

A factor is a number or variable that is multiplied to form a term. In 7x², the factors are 7, x, and x.

Coefficient

The coefficient is the numerical part multiplied by the variable(s). In 7x², “7” is the coefficient.

Example

In the expression 4ab + 3a² – 5b:

• Terms: 4ab, 3a², -5b
• Coefficients: 4 (for ab), 3 (for a²), -5 (for b)
• Variables: a, b
• Factors of 4ab: 4, a, b

3. Types of Algebraic Expressions

Algebraic expressions can be categorized based on the number of terms they have:

• Monomial: An expression with only one term. Example: 5x, -3ab²
• Binomial: An expression with two terms. Example: x + 7, 2a – 3b
• Trinomial: An expression with three terms. Example: a² + 2a + 1
• Polynomial: An expression with one or more terms. Example: 2x³ – 4x² + x – 5

Degree of an Expression

The degree of an algebraic expression is the highest sum of exponents of the variables in any term. For example, in 4x³y² + 3xy – 2, the degree is 5 (from x³y²).

4. Addition and Subtraction of Algebraic Expressions

Like and Unlike Terms

• Like Terms: Terms that have the same variables with the same exponents. Example: 5xy and –2xy are like terms.
• Unlike Terms: Terms with different variables or exponents. Example: 3x and 4y are unlike terms.

Addition/Subtraction Rule

Only like terms can be added or subtracted. Combine their coefficients.

Examples

• (3x + 5y) + (4x – 2y) = (3x + 4x) + (5y – 2y) = 7x + 3y
• (6a² – 2ab + b) – (2a² + 5b) = (6a² – 2a²) – 2ab + (b – 5b) = 4a² – 2ab – 4b

Real-World Example

If the revenue from ticket sales is 8x (for x students) and sponsorship is 3,500 rupees, the total income is expressed as 8x + 3,500.

5. Multiplication of Algebraic Expressions

Rules

• Multiply the coefficients together.
• Multiply the variables by adding their exponents (if the variables are the same).

Multiplying Monomials

Example: (3a) × (4a²) = 12a³

Multiplying a Monomial and a Binomial

Use the distributive law: a(b + c) = ab + ac.
Example: 2x(x + 5) = 2x² + 10x

Multiplying Binomials

Use the FOIL method (First, Outside, Inside, Last):
(x + 3)(x + 5) = x² + 5x + 3x + 15 = x² + 8x + 15

Multiplying Polynomials

Multiply each term in the first polynomial by every term in the second polynomial, then combine like terms.
Example: (x + 2)(x² + x + 1) = x³ + x² + x + 2x² + 2x + 2 = x³ + 3x² + 3x + 2

6. Special Products and Algebraic Identities

What is an Identity?

An algebraic identity is an equation that is true for all values of the variables. Identities help in quickly expanding expressions and simplifying complex problems.

Standard Algebraic Identities

• (a + b)² = a² + 2ab + b²
• (a – b)² = a² – 2ab + b²
• (a + b)(a – b) = a² – b²
• (x + a)(x + b) = x² + (a + b)x + ab
• (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca
• (a + b)³ = a³ + 3a²b + 3ab² + b³
• (a – b)³ = a³ – 3a²b + 3ab² – b³
• a³ + b³ = (a + b)(a² – ab + b²)
• a³ – b³ = (a – b)(a² + ab + b²)

Using Identities: Examples

• Expand (x + 7)² using (a + b)²:
  (x + 7)² = x² + 2 × x × 7 + 7² = x² + 14x + 49
• Expand (2a – 3b)²:
  (2a – 3b)² = (2a)² – 2 × 2a × 3b + (3b)² = 4a² – 12ab + 9b²
• Find the value of (17)² – (13)²:
  Using (a² – b²) = (a – b)(a + b):
  (17 – 13) × (17 + 13) = 4 × 30 = 120

7. Factorization of Algebraic Expressions

Factorization means expressing an algebraic expression as a product of its factors. This is often the reverse of expansion and is key in solving equations.

Common Methods of Factorization

• Taking out the common factor: ab + ac = a(b + c)
• Using identities: a² – b² = (a + b)(a – b)
• Grouping: x² + 5x + 6 = (x + 2)(x + 3)

Examples

• Factorize 6x² + 9x = 3x(2x + 3)
• Factorize x² – 16 = (x + 4)(x – 4)
• Factorize x² + 3x + 2 = (x + 1)(x + 2)

8. Applications and Word Problems

Practical Applications

• Geometry: Expressions for area and perimeter, e.g., the area of a rectangle: length × width = lw.
• Physics: Formulas like s = ut + ½at² involve algebraic expressions.
• Economics: Calculating profit, cost, and revenue.
• Everyday Life: Modeling patterns, budgeting, and problem-solving.

Example Word Problem

The length of a rectangular garden is (x + 5) meters, and the width is (x – 2) meters. Write an expression for the area.
Solution: Area = length × width = (x + 5)(x – 2) = x² + 3x – 10

Another Example

The sum of two numbers is y and their product is 12. Write an algebraic equation.
Solution: Let the numbers be a and b. a + b = y, ab = 12.

9. Common Errors and Misconceptions

• Adding unlike terms (e.g., 2x + 3y is not 5xy).
• Misapplying identities (e.g., (a + b)² ≠ a² + b²).
• Omitting coefficients or variable powers during multiplication.
• Failing to use the distributive property correctly.
• Ignoring the importance of order when subtracting binomials.
• Missing negative signs during expansion or factorization.

10. Practice Problems

Basic

1. Identify variables, constants, and coefficients in 5x² – 7x + 9.
2. Simplify: 2a + 3a – 4a + 5.
3. Add: (2x + 3y) + (4x – y).
4. Multiply: (x + 2)(x + 5).
5. Expand: (a – 4)².

Intermediate

6. Factorize: x² – 6x + 9.
7. Simplify: 7m – 3n + 4m + 2n.
8. Expand: (2x + 3)(x – 1).
9. If the area of a square is x² + 8x + 16, find the side.
10. Simplify: (x² – 2x + 1) + (3x² + 5x – 7).

Advanced

11. Use identities to expand: (2x + 3y)².
12. Factorize: a³ – b³.
13. If the perimeter of a rectangle is 2x + 2y and its area is xy, write expressions for length and width in terms of x and y.
14. Simplify: (x + 4)(x – 4) – x².
15. Find the product: (a + b + c)(a + b – c).

11. Solutions

1. Variables: x; Constants: 9; Coefficients: 5 (for x²), –7 (for x).
2. 2a + 3a – 4a + 5 = (2a + 3a – 4a) + 5 = (5a – 4a) + 5 = a + 5.
3. (2x + 3y) + (4x – y) = (2x + 4x) + (3y – y) = 6x + 2y.
4. (x + 2)(x + 5) = x² + 5x + 2x + 10 = x² + 7x + 10.
5. (a – 4)² = a² – 8a + 16.
6. x² – 6x + 9 = (x – 3)².
7. 7m – 3n + 4m + 2n = (7m + 4m) + (–3n + 2n) = 11m – n.
8. (2x + 3)(x – 1) = 2x² + 3x – 2x – 3 = 2x² + x – 3.
9. x² + 8x + 16 = (x + 4)² ⇒ Side = x + 4.
10. (x² – 2x + 1) + (3x² + 5x – 7) = (x² + 3x²) + (–2x + 5x) + (1 – 7) = 4x² + 3x – 6.
11. (2x + 3y)² = (2x)² + 2×2x×3y + (3y)² = 4x² + 12xy + 9y².
12. a³ – b³ = (a – b)(a² + ab + b²).
13. Perimeter = 2x + 2y ⇒ length = x, width = y. Area = xy.
14. (x + 4)(x – 4) – x² = (x² – 16) – x² = –16.
15. (a + b + c)(a + b – c) = (a + b)² – c² = a² + 2ab + b² – c².

Conclusion

Algebraic expressions and identities are foundational for algebra and many branches of mathematics. By learning to recognize, manipulate, expand, factorize, and apply expressions and identities, you unlock the power to solve real-world and abstract problems with confidence. Keep practicing these skills and apply them to diverse situations—from geometry and science to finance and beyond!