Algebra Basics Lecture Notes

Key Concepts

• Coefficient: The number in front of a variable in an algebraic expression. For example, in 3x^2, the coefficient is 3.
• Variable: The letter representing a number that can change. In 3x^2, x is the variable.
• Exponent: The power to which the variable is raised. In 3x^2, the exponent is 2. Also known as an indice.

Types of Algebraic Expressions

• Monomial: An expression with one term, e.g., 5x.
• Binomial: An expression with two terms, e.g., 4x + 3.
• Trinomial: An expression with three terms, e.g., 3x^2 + 2x + 1.
• Polynomial: An expression with more than three terms. "Poly" means many.

Identifying Parts of an Expression

• Terms Counting: Count the distinct parts separated by + or - signs.
• Coefficient of a Term: It's the number directly in front of each term (include its sign).
• Constant Term: A term without a variable (e.g., +2 in 3x^2 + 4x + 2 is constant).

Ordering Terms by Exponents

• Descending Order: Arrange terms from largest to smallest exponent.
• Ascending Order: Arrange terms from smallest to largest exponent.

Solved Examples

1. Given Expression: 5x^3 - 6x^2 + 4x - 1

   • Count terms: 4 (Polynomial)
   • Arrange in descending order: 5x^3, -6x^2, 4x, -1
   • Coefficient of x^2 term: -6

2. Another Expression: 2x - 8x^2 + 3x^3 + 7

   • Arrange in ascending order: 2x, -8x^2, 3x^3, 7
   • Count terms: 4 (Polynomial)
   • Coefficient of the x^2 term: -8

Additional Notes

• Always include signs with coefficients.
• Remember, if there seems to be no visible coefficient, it is 1.
• Zero exponent means the term is a constant (e.g., x^0 = 1).
• Practice arranging terms in both ascending and descending order to solidify understanding.