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.

Learn about the definition and characteristics of polynomial functions, including the domain and range. Test your understanding of polynomial equations and expressions.