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 n is a non-negative integer and a_n ≠ 0.

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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 f(x) = 2x^3 - 3x^2 + 4x - 10, the degree is 3.

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What is the maximum number of turning points in a polynomial function?

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The maximum number of turning points in a polynomial function is given by the formula: maximum turning points = degree - 1.

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What is end behavior in polynomial functions?

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End behavior describes the behavior of the graph of a polynomial function as x approaches positive or negative infinity. It is determined by the leading term of the polynomial.

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

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To determine the end behavior, look at the leading term of the polynomial. If the leading coefficient is positive and the degree is even, the ends go up. If the leading coefficient is negative and the degree is even, the ends go down.

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What is a zero of a polynomial function?

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A zero of a polynomial function is a value of x for which f(x) = 0. It represents the x-intercepts of the graph.

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How many zeros can a polynomial function have?

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A polynomial function of degree n can have at most n real zeros.

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What is a relative extremum?

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A relative extremum is a point on the graph of a function where the function takes on a maximum or minimum value relative to nearby points.

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What is the relationship between turning points and relative extrema?

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Turning points are points where the graph changes direction, and they can correspond to relative extrema. A polynomial function can have at most n-1 turning points, where n is the degree of the polynomial.

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What is the significance of the leading coefficient in a polynomial function?

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The leading coefficient determines the direction of the graph's end behavior. If it is positive, the graph rises to the right; if negative, it falls to the right.