Algebra II Polynomials Flashcards

Key Polynomial Concepts

• Monomial: A single term expression in algebra.
• Binomial: An expression consisting of two terms.
• Trinomial: An expression with three terms.
• Polynomial: An expression with more than three terms.

Polynomial Forms

• Standard Form: Arrangement of polynomial terms from the largest to smallest exponent.
• Factored Form: Expressed as a product of factors, e.g., f(x) = (x-3)(x+2)(x+1).

Polynomial Characteristics

• Degree: The highest exponent in standard form; in factored form, it is the sum of the multiplicities of the factors.
• Leading Coefficient: The coefficient of the term with the highest degree, important for end behavior.

Roots and Zeros

• Zeros: The x-intercepts or solutions of the polynomial equation.
• Multiplicity: Indicates how many times a factor appears, affecting how the graph interacts with the x-axis at zeros.

Graph Analysis

• End Behavior: Determined by the degree (even for parabolas, odd for snake-like behavior) and the sign of the leading coefficient.
• Relative Maximum: The point where the graph changes from increasing to decreasing.
• Relative Minimum: The point where the graph transitions from decreasing to increasing.

Polynomial Division

• Dividend: The expression or number under the division sign.
• Divisor: The expression or number outside the division sign.
• Quotient: The result obtained from division.
• Synthetic Division: A method used for polynomials, particularly when the polynomial is not easily factorable.

Solving Polynomials

• Rational Root Theorem: A technique for finding rational zeros in polynomials with 4 or more terms that cannot be factored by grouping.
• Graphing: Requires factoring the polynomial to find x-intercepts for visualization.
• Synthetic Division Use: Necessary for polynomials that cannot be solved by simple factorization.

Problem-Solving Insights

• Degree Determination: In standard form, it's the highest exponent; in factored form, sum the multiplicities.
• Synthetic Division Solution Checking: A remainder of zero indicates a solution; otherwise, it’s not a solution.
• Quadratic Formula Limitation: Not applicable to non-trinomials, such as cubic equations, which have a degree of 3.

Example Functions

• Functions exhibiting two relative minimums and one maximum can have a positive leading coefficient and an even degree, such as 4.
• Some polynomials require synthetic division due to non-factorable grouping, typically those with 4 or more terms.

Additional Considerations

• Not all solutions are found using the Rational Root Theorem, as it does not account for imaginary or irrational roots.

Test your understanding of Polynomials in Algebra II with these flashcards. Each card covers key terms such as Monomial, Standard Form, and Rational Root Theorem, helping you prepare effectively for your test.