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Questions and Answers
What is the general form of a polynomial function?
What is the general form of a polynomial function?
What is the degree of a polynomial function?
What is the degree of a polynomial function?
What is the range of a polynomial function?
What is the range of a polynomial function?
What is a monomial?
What is a monomial?
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How do you add or subtract polynomial functions?
How do you add or subtract polynomial functions?
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What is the shape of the graph of a polynomial function?
What is the shape of the graph of a polynomial function?
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What are the x-intercepts of a polynomial function?
What are the x-intercepts of a polynomial function?
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What is an application of polynomial functions?
What is an application of polynomial functions?
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Study Notes
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.
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Description
Learn about the definition and characteristics of polynomial functions, including the domain and range. Test your understanding of polynomial equations and expressions.
