Algebra Chapter: Definitions and Types of Equations

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Questions and Answers

What is the definition of a variable in algebra?

  • A symbol that represents a fixed value
  • A symbol that represents a value that can change (correct)
  • A combination of mathematical operations
  • A statement that says two expressions are equal

What is the highest power of the variable in a linear equation?

  • 0
  • 3
  • 1 (correct)
  • 2

How can quadratic equations be solved?

  • Using only addition and subtraction
  • Using synthetic division
  • Using only multiplication and division
  • Using factoring, quadratic formula, or completing the square (correct)

What is the formula to find the nth term of an arithmetic progression?

<p>an = a1 + (n - 1)d (D)</p> Signup and view all the answers

What is the sum of the first n terms of an arithmetic progression?

<p>Sn = n/2 [2a1 + (n - 1)d] (D)</p> Signup and view all the answers

What is the 10th term of the arithmetic progression 3, 6, 9, 12,...?

<p>36 (D)</p> Signup and view all the answers

What is the common property of all arithmetic progressions?

<p>The common difference is constant (C)</p> Signup and view all the answers

What type of equation is x^3 - 2x^2 + x + 1 = 0?

<p>Polynomial equation (C)</p> Signup and view all the answers

What is the degree of the polynomial equation x^3 - 2x^2 + x + 1 = 0?

<p>3 (D)</p> Signup and view all the answers

What is the characteristic of an arithmetic progression that distinguishes it from other types of sequences?

<p>The common difference is constant (B)</p> Signup and view all the answers

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Study Notes

Algebra

Definitions

  • Algebra: a branch of mathematics that deals with variables and their relationships
  • Variable: a symbol that represents a value that can change
  • Constant: a symbol that represents a fixed value
  • Expression: a combination of variables, constants, and mathematical operations
  • Equation: a statement that says two expressions are equal

Types of Equations

  • Linear equation: an equation in which the highest power of the variable is 1
    • Example: 2x + 3 = 5
  • Quadratic equation: an equation in which the highest power of the variable is 2
    • Example: x^2 + 4x + 4 = 0
  • Polynomial equation: an equation consisting of variables and coefficients combined using only addition, subtraction, and multiplication
    • Example: x^3 - 2x^2 + x + 1 = 0

Solving Equations

  • Linear equations: can be solved using addition, subtraction, multiplication, and division
  • Quadratic equations: can be solved using factoring, quadratic formula, or completing the square
  • Polynomial equations: can be solved using factoring, remainder theorem, or synthetic division

Arithmetic Progression

Definition

  • Arithmetic progression (AP): a sequence of numbers in which each term is obtained by adding a fixed constant to the previous term
  • Common difference (d): the fixed constant added to each term to get the next term

Formulae

  • nth term of an AP: an = a1 + (n - 1)d
  • Sum of first n terms of an AP: Sn = n/2 [2a1 + (n - 1)d]

Properties

  • The common difference is constant
  • The sequence is infinite
  • The terms can be positive, negative, or zero

Examples

  • 2, 5, 8, 11, ... is an AP with a1 = 2 and d = 3
  • Find the 10th term of the AP: 3, 6, 9, 12, ...

Algebra

Types of Equations

  • A linear equation has the highest power of the variable as 1, e.g., 2x + 3 = 5
  • A quadratic equation has the highest power of the variable as 2, e.g., x^2 + 4x + 4 = 0
  • A polynomial equation consists of variables and coefficients combined using addition, subtraction, and multiplication, e.g., x^3 - 2x^2 + x + 1 = 0

Solving Equations

  • Linear equations can be solved using addition, subtraction, multiplication, and division
  • Quadratic equations can be solved using factoring, quadratic formula, or completing the square
  • Polynomial equations can be solved using factoring, remainder theorem, or synthetic division

Arithmetic Progression

Key Concepts

  • An arithmetic progression (AP) is a sequence of numbers where each term is obtained by adding a fixed constant to the previous term
  • The fixed constant is called the common difference (d)
  • The sequence is infinite, and terms can be positive, negative, or zero

Formulae

  • The nth term of an AP is an = a1 + (n - 1)d
  • The sum of first n terms of an AP is Sn = n/2 [2a1 + (n - 1)d]

Examples and Applications

  • The sequence 2, 5, 8, 11,...is an AP with a1 = 2 and d = 3
  • To find the 10th term of the AP: 3, 6, 9, 12,..., use the formula an = a1 + (n - 1)d

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