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_n*x^n + a_(n-1)*x^(n-1) + ... + a_1*x + a_0, where a_n ≠ 0.

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What are the degrees of polynomial functions?

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The degree of a polynomial function is the highest power of the variable in the expression. It determines the shape and behavior of the graph.

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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 influences the end behavior of the graph.

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How do you find the relative extrema of a polynomial function?

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Relative extrema can be found by taking the derivative of the polynomial, setting it to zero, and solving for x. The solutions indicate critical points where the function may have local maxima or minima.

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What is the factored form of a polynomial?

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The factored form of a polynomial expresses it as a product of its linear factors. For example, f(x) = (x - r1)(x - r2)...(x - rn) where r1, r2,..., rn are the roots of the polynomial.

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What is the standard form of a polynomial?

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The standard form of a polynomial is written in descending order of the degree of its terms. For example, f(x) = ax^n + bx^(n-1) + ... + c.

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What does it mean for a polynomial to have an even degree?

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A polynomial has an even degree if the highest power of the variable is an even number. This typically results in a graph that opens in the same direction at both ends.

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What does it mean for a polynomial to have a positive leading coefficient?

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A polynomial has a positive leading coefficient if the coefficient of the term with the highest degree is positive. This means the graph will rise to the right.

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What is the significance of the roots of a polynomial?

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The roots of a polynomial are the values of x that make the polynomial equal to zero. They correspond to the x-intercepts of the graph.

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How do you determine the end behavior of a polynomial function?

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The end behavior of a polynomial function is determined by its degree and leading coefficient. For even degrees, both ends behave the same; for odd degrees, they behave oppositely.