Algebra: Polynomials and Terminology

Definition and Terminology

• A polynomial is an expression consisting of variables (such as x or y) and coefficients (constants) combined using only addition, subtraction, and multiplication.
• Polynomials can be classified based on the degree (highest power of the variable):
  • Monomials: polynomials with one term
  • Binomials: polynomials with two terms
  • Trinomials: polynomials with three terms
• Degree of a polynomial: the highest power of the variable
• Leading coefficient: the coefficient of the term with the highest degree

Operations with Polynomials

• Adding and subtracting polynomials:
  • Combine like terms (terms with the same variable and degree)
  • Simplify the expression
• Multiplying polynomials:
  • Distribute each term in one polynomial to each term in the other polynomial
  • Combine like terms
• Factoring polynomials:
  • Express a polynomial as a product of simpler expressions
  • Common factors, difference of squares, and sum and difference formulas can be used

Properties of Polynomials

• Commutative property: the order of the variables does not change the polynomial
• Associative property: the order in which terms are added or multiplied does not change the polynomial
• Distributive property: multiplication distributes over addition

Graphs of Polynomials

• The graph of a polynomial is a smooth curve
• The degree of a polynomial determines the shape of the graph:
  • Even degree: graph has a minimum or maximum point
  • Odd degree: graph has a point of inflection
• x-intercepts: points where the graph crosses the x-axis
• y-intercept: point where the graph crosses the y-axis

Applications of Polynomials

• Modeling real-world phenomena, such as population growth or electrical circuits
• Approximating functions, such as trigonometric or exponential functions
• Solving equations and inequalities

Definition and Terminology

• A polynomial is an expression consisting of variables and coefficients combined using addition, subtraction, and multiplication.
• Polynomials can be classified based on the degree (highest power of the variable).
• Monomials are polynomials with one term.
• Binomials are polynomials with two terms.
• Trinomials are polynomials with three terms.
• The degree of a polynomial is the highest power of the variable.
• The leading coefficient is the coefficient of the term with the highest degree.

Operations with Polynomials

• When adding or subtracting polynomials, combine like terms and simplify the expression.
• When multiplying polynomials, distribute each term in one polynomial to each term in the other polynomial and combine like terms.
• Factoring polynomials involves expressing a polynomial as a product of simpler expressions, using common factors, difference of squares, and sum and difference formulas.

Properties of Polynomials

• The commutative property states that the order of the variables does not change the polynomial.
• The associative property states that the order in which terms are added or multiplied does not change the polynomial.
• The distributive property states that multiplication distributes over addition.

Graphs of Polynomials

• The graph of a polynomial is a smooth curve.
• The degree of a polynomial determines the shape of the graph: even degree graphs have a minimum or maximum point, while odd degree graphs have a point of inflection.
• x-intercepts are points where the graph crosses the x-axis.
• The y-intercept is the point where the graph crosses the y-axis.

Applications of Polynomials

• Polynomials can model real-world phenomena, such as population growth or electrical circuits.
• Polynomials can approximate functions, such as trigonometric or exponential functions.
• Polynomials can be used to solve equations and inequalities.