Questions and Answers
What is a monomial?
What is a binomial?
What defines a trinomial?
What is a polynomial?
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What is the degree of a polynomial?
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Define a polynomial function.
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What is the standard form of a polynomial function?
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What is a turning point?
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What does end behavior refer to?
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What does the Factor Theorem state?
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Define a multiple zero.
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What is multiplicity in relation to polynomials?
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What is a relative maximum?
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What is a relative minimum?
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What is the sum of cubes formula?
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State the difference of cubes formula.
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What is a perfect square trinomial?
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What is the formula for the difference of two squares?
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Explain factor by grouping.
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What does the Conjugate Root Theorem state?
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What is Descartes' Rule of Signs?
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Define the Rational Root Theorem.
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What is the Remainder Theorem?
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What is synthetic division?
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Study Notes
Polynomial Functions Overview
- Monomial: A single term consisting of a real number, variable, or a product with whole-number exponents (e.g. 4x⁴yz²).
- Binomial: Sum or difference of two monomials (e.g. 3x²+5xy⁴).
- Trinomial: Sum or difference of three monomials (e.g. x⁴y⁶+x³y⁵+x²z).
- Polynomial: A monomial or sum of monomials.
Polynomial Characteristics
- Degree: The highest exponent in a polynomial, indicating its overall degree.
- Polynomial Function: An equation that includes polynomials.
- Standard Form: Polynomials expressed from highest to lowest degree in the form P(x) = 4x³+3x²+5x-2.
Graphing Polynomials
- Turning Point: Locations on the graph where the polynomial changes direction.
- End Behavior: Describes how the graph behaves as x approaches positive or negative infinity.
Key Theorems and Concepts
- Factor Theorem: A polynomial includes (x−a) as a factor if a is a zero of the polynomial's function.
- Multiple Zero: The occurrence of a zero from a repeated linear factor.
- Multiplicity: The exponent that indicates how many times a linear factor is repeated.
Analyzing Polynomials
- Relative Maximum: The function value at a turning point where the graph ascends then descends.
- Relative Minimum: The function value at a turning point where the graph descends then ascends.
Special Products
- Sum of Cubes: a³ + b³ = (a + b)(a² - ab + b²).
- Difference of Cubes: a³ - b³ = (a - b)(a² + ab + b²).
- Perfect Square Trinomial: a² + 2ab + b² = (a + b)²; a² - 2ab + b² = (a - b)².
- Difference of Two Squares: a² - b² = (a - b)(a + b).
Factoring Techniques
- Factor by Grouping: A method to factor quadratics where terms are grouped (e.g., a(x+y) + b(x+y) = (a+b)(x+y)).
Additional Theorems
- Conjugate Root Theorem: Complex roots appear in conjugate pairs, indicating if a + √b is a root, then a - √b is also a root.
- Descartes' Rule of Signs: Analyzes a polynomial's sign changes to predict the number of positive and negative real roots.
- Rational Root Theorem: Identifies possible rational roots based on factors of the constant term and leading coefficient.
- Remainder Theorem: Simplifies finding the remainder of polynomial division; P(a) gives the remainder when dividing by (x-a).
- Synthetic Division: A streamlined method for dividing polynomials by linear factors.
These notes provide a comprehensive overview of polynomial functions, their properties, factoring techniques, and theorems useful in analysis and problem solving.
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Test your knowledge of polynomial functions with these flashcards. Each card presents a key term along with its definition, providing a comprehensive overview of important concepts in Algebra 2. Ideal for reviewing monomials, binomials, trinomials, and polynomials.