Algebra: Polynomial Functions

Definition

• A polynomial function is a function composed of variables and coefficients combined using only addition, subtraction, and multiplication.
• It can be expressed in the form: f(x) = anx^n + an-1x^(n-1) + … + a1*x + a0
• where:
  • an is the leading coefficient (non-zero)
  • n is the degree of the polynomial (highest power of x)
  • a0 is the constant term

Characteristics

• Domain: The set of all real numbers (unless specified otherwise)
• Range: Depends on the polynomial; may include all real numbers
• Continuity: Polynomial functions are continuous for all real numbers
• Differentiability: Polynomial functions are differentiable for all real numbers

Classification

• Monomials: Polynomials with one term (e.g., 3x^2)
• Binomials: Polynomials with two terms (e.g., x^2 + 3x)
• Trinomials: Polynomials with three terms (e.g., x^2 + 2x + 1)
• Monic: Polynomials with leading coefficient 1 (e.g., x^2 + 2x + 1)

Operations

• Addition: Combine like terms
• Subtraction: Combine like terms
• Multiplication: Distribute each term in one polynomial to each term in the other
• Division: Divide each term in the dividend by the divisor (not always possible)

Graphical Representation

• Shape: The graph of a polynomial function can be a parabola, a cubic curve, or a higher-degree curve
• Intercepts: x-intercepts are the roots of the polynomial (values of x that make the function equal to zero)
• Asymptotes: The graph may have vertical asymptotes, but not horizontal asymptotes (since polynomials grow without bound)

Applications

• Modeling: Polynomial functions can model various real-world phenomena, such as population growth, motion, and electrical circuits
• Algebra: Polynomials are used to solve equations and inequalities
• Calculus: Polynomials are used to approximate more complex functions in calculus

Definition of Polynomial Functions

• A polynomial function is composed of variables and coefficients combined using addition, subtraction, and multiplication.
• The standard form of a polynomial function is: f(x) = anx^n + an-1x^(n-1) + … + a1*x + a0.
• an is the leading coefficient (non-zero), n is the degree of the polynomial (highest power of x), and a0 is the constant term.

Characteristics of Polynomial Functions

• The domain of polynomial functions is the set of all real numbers, unless specified otherwise.
• The range of polynomial functions depends on the polynomial, and may include all real numbers.
• Polynomial functions are continuous and differentiable for all real numbers.

Classification of Polynomials

• Monomials are polynomials with one term, such as 3x^2.
• Binomials are polynomials with two terms, such as x^2 + 3x.
• Trinomials are polynomials with three terms, such as x^2 + 2x + 1.
• Monic polynomials have a leading coefficient of 1, such as x^2 + 2x + 1.

Operations with Polynomials

• To add or subtract polynomials, combine like terms.
• To multiply polynomials, distribute each term in one polynomial to each term in the other.
• Division of polynomials is possible, but not always; divide each term in the dividend by the divisor.

Graphical Representation of Polynomials

• The graph of a polynomial function can be a parabola, a cubic curve, or a higher-degree curve.
• x-intercepts are the roots of the polynomial, or values of x that make the function equal to zero.
• The graph of a polynomial function may have vertical asymptotes, but not horizontal asymptotes.

Applications of Polynomial Functions

• Polynomial functions can model real-world phenomena, such as population growth, motion, and electrical circuits.
• Polynomials are used to solve equations and inequalities in algebra.
• Polynomials are used to approximate more complex functions in calculus.