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: $$f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0$$ where $$a_n, a_{n-1}, ..., a_0$$ are constants and $$n$$ is a non-negative integer.

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What is the degree of a polynomial?

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The degree of a polynomial is the highest power of the variable in the polynomial expression. For example, in the polynomial $$f(x) = 2x^3 - 4x^2 + 3$$, the degree is 3.

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What does it mean for a polynomial to be even?

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A polynomial is even if it satisfies the condition $$f(-x) = f(x)$$ for all $$x$$ in its domain. This means the graph of the polynomial is symmetric about the y-axis.

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What does it mean for a polynomial to be odd?

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A polynomial is odd if it satisfies the condition $$f(-x) = -f(x)$$ for all $$x$$ in its domain. This means the graph of the polynomial is symmetric about the origin.

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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. For example, in $$f(x) = 5x^4 + 3x^2 - 2$$, the leading coefficient is 5.

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What are the roots of a polynomial?

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The roots of a polynomial are the values of $$x$$ for which the polynomial equals zero. They can be found by solving the equation $$f(x) = 0$$.

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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 $$n$$ roots in the complex number system, counting multiplicities.

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What is a polynomial's end behavior?

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The end behavior of a polynomial describes how the function behaves as $$x$$ approaches positive or negative infinity. It is determined by the leading term.

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What is a discontinuity in a polynomial function?

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A discontinuity in a polynomial function is a point at which the function is not continuous. Polynomials are continuous everywhere, so they have no discontinuities.

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What is the difference between a polynomial and a rational function?

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A polynomial is a function that involves only non-negative integer powers of $$x$$, while a rational function is a ratio of two polynomials.