Algebra 2 - Polynomial Functions Flashcards

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Questions and Answers

What is a monomial?

A polynomial arranged by degree in descending order

Three monomials connected with addition or subtraction

A real number, variable or product of a real number and one or more variables with whole-number exponents (correct)

Two monomials connected with addition or subtraction

What is a binomial?

A real number

Three monomials connected with addition or subtraction

Two monomials connected with addition or subtraction (correct)

A monomial or a sum of monomials

What defines a trinomial?

Three monomials connected with addition or subtraction (correct)

Two monomials connected with addition or subtraction

Any expression with more than three terms

A single term

What is a polynomial?

A monomial or a sum of monomials Signup and view all the answers

What is the degree of a polynomial?

The value of the greatest exponent for a polynomial in one variable Signup and view all the answers

Define a polynomial function.

An equation with a polynomial Signup and view all the answers

What is the standard form of a polynomial function?

A polynomial arranged by degree in descending numerical order Signup and view all the answers

What is a turning point?

Place where the graph of a polynomial changes direction Signup and view all the answers

What does end behavior refer to?

Direction of the graph to the far left and the far right Signup and view all the answers

What does the Factor Theorem state?

The expression x−a is a factor of a polynomial if and only if a is a zero of the related polynomial function Signup and view all the answers

Define a multiple zero.

The zero of a repeated linear factor Signup and view all the answers

What is multiplicity in relation to polynomials?

The exponent of a repeated linear factor Signup and view all the answers

What is a relative maximum?

The value of a function at an up-to-down turning point Signup and view all the answers

What is a relative minimum?

The value of a function at a down-to-up turning point Signup and view all the answers

What is the sum of cubes formula?

a³ + b³ = (a + b)(a² - ab + b²) Signup and view all the answers

State the difference of cubes formula.

a³ - b³ = (a - b)(a² + ab + b²) Signup and view all the answers

What is a perfect square trinomial?

a² + 2ab + b² = (a + b)² or a² - 2ab + b² = (a - b)² Signup and view all the answers

What is the formula for the difference of two squares?

a² - b² = (a - b)(a + b) Signup and view all the answers

Explain factor by grouping.

ax + ay + bx + by = a(x + y) + b(x + y) = (a + b)(x + y) Signup and view all the answers

What does the Conjugate Root Theorem state?

Complex roots occur in conjugate pairs Signup and view all the answers

What is Descartes' Rule of Signs?

It helps determine the possible number of positive or negative real roots in a polynomial Signup and view all the answers

Define the Rational Root Theorem.

There are a limited number of rational roots of any polynomial Signup and view all the answers

What is the Remainder Theorem?

A quick way to find the remainder of a polynomial long-division problem Signup and view all the answers

What is synthetic division?

A method that makes polynomial division easier when dividing by a linear factor Signup and view all the answers

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.