Algebra Class: Expressions and Polynomials

Solving Algebraic Expressions

• Algebraic expressions are mathematical phrases involving variables, constants, and operations (+, -, ×, ÷).
• To solve an algebraic expression, simplify by combining like terms.
  • Like terms have the same variable(s) raised to the same power(s).
• Example: Simplify 3x + 5y - 2x + y.
  • Combine the x terms: 3x - 2x = x
  • Combine the y terms: 5y + y = 6y
  • The simplified expression is x + 6y.
• Solving may involve evaluating the expression for specific variable values.

Adding and Subtracting Polynomials

• A polynomial is an expression with variables and coefficients, using addition, subtraction, multiplication, and non-negative integer exponents.
• To add or subtract polynomials, combine like terms.
• Example: Add (2x² + 3x - 1) and (x² - 5x + 4).
  • Group like terms: (2x² + x²) + (3x - 5x) + (-1 + 4)
  • Combine like terms: 3x² - 2x + 3
• When subtracting, distribute the negative to every term in the second polynomial.

Multiplication of Polynomials

• Multiplying polynomials involves distributing each term in one polynomial to each term in the other.
  • Use the distributive property. (FOIL is a specific method for binomials).
• Example: Multiply (x + 2) and (x + 3)
  • (x + 2)(x + 3) = x(x + 3) + 2(x + 3)
  • = x² + 3x + 2x + 6
  • = x² + 5x + 6
• Multiplying any number of polynomials requires consistent distribution.

Division of Polynomials

• Polynomial division uses long division, similar to numerical long division.
• Example: Divide 2x³ + 5x² - x + 3 by (x + 1).
  • Set up division with terms in descending order.
• Polynomial division results in a quotient and a remainder (potentially zero).

Finding the Degree of Polynomials

• The degree of a polynomial is the highest power of the variable in the expression.
• Example: The degree of 4x³ + 2x² - x + 5 is 3.
  • The highest power of x is 3.
• The degree helps understand the polynomial's behavior.
• Zero terms are simplified; the highest exponent determines the degree.