Fundamental Concepts of Algebra Chapter Notes - Class 6 PDF Download

Introduction

Algebra is like a magical language in mathematics that helps us solve puzzles and find patterns using numbers, letters, and symbols. Imagine you're a detective trying to figure out how many candies you need for your friends or how many matchsticks make a cool pattern! Algebra uses tools like letters (a, b, x, y) and symbols (+, −, ×, ÷) to create general rules for such problems. The word "algebra" comes from an ancient book called 'Aljebar w'al almugabalah,' which means solving equations. In this chapter, we'll explore how algebra helps us describe patterns, calculate quantities, and express relationships in a fun and simple way!

Constants and Variables

• A constant is a number with a fixed value that never changes.
• Examples of constants include 2, -5, 10, and ³/₇.
• A variable is a letter or symbol whose value can change.
• Examples of variables include x, y, z, 6a, and 7b.
• Variables are also called literal numbers or literals.
• A combination of constants and variables (like 3x or 5ab) is treated as a variable because its value depends on the variable.
• All rules of addition, subtraction, multiplication, and division for numbers apply to literals too.

Matchstick Patterns

• Matchstick patterns help us find general rules for the number of matchsticks needed to form shapes like squares or triangles.
• We observe the pattern, create a table, and use a variable to write a formula.
• Formula for squares: Number of matchsticks = 3n + 1, where n is the number of squares.
• Formula for triangles: Number of matchsticks = 3n, where n is the number of triangles.

Use of Variables

• Variables are used to write general formulas for real-life situations or geometric properties.
• They help express quantities that depend on other factors, like the number of items or measurements.

Perimeter of a Square

• The perimeter of a square is the total length of its four sides.
• Formula: Perimeter = 4 × side length = 4l, where l is the side length.

Area of a Rectangle

• The area of a rectangle is the product of its length and breadth.
• Formula: Area = length × breadth = l × b, where l is length and b is breadth.
• Here, l and b are variables that can take any positive value.

Total Length of Edges of a Cube

• A cube has 12 edges, all of equal length.
• Formula: Total length of edges = 12 × edge length = 12l, where l is the edge length.

Terms

• A term is a single constant, variable, or a product/quotient of constants and variables.
• Examples: 5, 15ab, ⁷/_y are terms.
• A constant term has no variable, e.g., 5 or -3.
• In a term, the constant part is the numerical factor, and the variable part is the literal factor.
• Example: In -4xy, -4 is the numerical factor, and x, y, xy are literal factors.

Like Terms

• Like terms have the same literal factors with the same powers.
• They differ only in their numerical factors.
• Example: 6xy and -7xy are like terms because they both have xy.

Unlike Terms

• Unlike terms have different literal factors or different powers.
• Example: 3x² and 3xy are unlike terms because their literal factors differ.

Coefficient

• A coefficient is any factor of a term when considered as a product of the remaining factors.
• In x²y, the coefficient of x² is y, and the coefficient of y is x².

Algebraic Expressions

• An algebraic expression is a combination of one or more terms joined by + or −.
• Examples:
•  ab (1 term), 
• 3xy + ^x/_a (2 terms), 
• 2x² + y² − ^x/_y (3 terms).
• Expressions can have variables with negative or fractional powers, but these are not polynomials.

Monomial

• A monomial is an algebraic expression with exactly one non-zero term and whole number powers of variables.
• Examples: a, xy, 5xy, ³/₇ab.

Binomial

• A binomial is an algebraic expression with exactly two non-zero terms and whole number powers of variables.
• Examples: 3x + a, 5x + 17y, a + b.

Trinomial

• A trinomial is an algebraic expression with exactly three non-zero terms and whole number powers of variables.
• Examples: x + y² + 5a, xy + z + 2ab.

Polynomial

• A polynomial is an algebraic expression with a finite number of non-zero terms and whole number powers of variables.
• All monomials, binomials, and trinomials with whole number powers are polynomials.
• Expressions with negative or fractional powers (like √x or ¹/_m) are not polynomials.

Degree of a Polynomial

• The degree of a polynomial is the highest power of the variable in any of its terms.
• A constant polynomial (like 7) has degree 0.

Solved Examples

Example 1: Find the rule for the number of matchsticks needed for a pattern of letter 'A'.
Solution: For n letters 'A', the number of matchsticks is 4n + 2. For 5 letters, 4 × 5 + 2 = 22 matchsticks.

Example 2: Sita is 7 years younger than Radha. Express Radha's age in terms of Sita's age.
Solution: Let Sita's age be x years. Radha's age = x + 7 years.
Example 3: Separate the following into pairs of like terms: 2xy, −4a, x²y, ¹/₃ab, a, 5ab, 6x²y, ⁻¹⁴/₁₅xy.
Solution: Pairs are: (2xy, ⁻¹⁴/₁₅xy), (−4a, a), (x²y, 6x²y), (¹/₃ab, 5ab).
Example 4: Find the coefficient of xy in −a²bxy.
Solution: The factors other than xy are −a² and b. Thus, the coefficient is −a²b.
Example 5: Classify −7xy + x² as monomial, binomial, or trinomial.
Solution: It has two terms (−7xy, x²), so it is a binomial.