Polynomial Functions

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What is a polynomial function?

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A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. The general form is P(x) = a_n*x^n + a_(n-1)*x^(n-1) + ... + a_1*x + a_0, where n is a non-negative integer.

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How do you classify polynomials by degree?

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Polynomials are classified by their degree, which is the highest power of the variable in the expression. For example, a polynomial of degree 0 is a constant, degree 1 is linear, degree 2 is quadratic, degree 3 is cubic, degree 4 is quartic, degree 5 is quintic, and so on.

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What is the leading coefficient of a polynomial?

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The leading coefficient is the coefficient of the term with the highest degree in a polynomial. It determines the end behavior of the polynomial function.

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What does the end behavior of a polynomial refer to?

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End behavior describes how the values of a polynomial function behave as the input (x) approaches positive or negative infinity. It is influenced by the degree and leading coefficient of the polynomial.

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How do you find the zeros of a polynomial function?

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To find the zeros of a polynomial function, set the polynomial equal to zero and solve for the variable. This can involve factoring, using the quadratic formula, or applying synthetic division.

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What is the Fundamental Theorem of Algebra?

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The Fundamental Theorem of Algebra states that every non-constant polynomial function of degree n has exactly n complex roots, counting multiplicities.

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What is a quartic polynomial?

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A quartic polynomial is a polynomial of degree 4, which can be expressed in the form P(x) = ax^4 + bx^3 + cx^2 + dx + e, where a ≠ 0.

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What is a quintic polynomial?

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A quintic polynomial is a polynomial of degree 5, expressed as P(x) = ax^5 + bx^4 + cx^3 + dx^2 + ex + f, where a ≠ 0.

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What is the difference between even and odd degree polynomials?

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Even degree polynomials have the same end behavior on both sides (both rise or both fall), while odd degree polynomials have opposite end behavior (one side rises and the other falls).

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How do you solve the equation x^4 - 256 = 0?

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To solve x^4 - 256 = 0, factor it as (x^2 - 16)(x^2 + 16) = 0. This gives x^2 - 16 = 0 (x = 4, x = -4) and x^2 + 16 = 0 (x = 4i, x = -4i).