Cubic Functions

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What is a cubic function?

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A cubic function is a polynomial function of degree three, typically expressed in the form f(x) = ax^3 + bx^2 + cx + d, where a, b, c, and d are constants and a ≠ 0.

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What does it mean if x = r is a zero of f(x)?

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If x = r is a zero of f(x), it means that f(r) = 0, indicating that the graph of the function intersects the x-axis at the point (r, 0).

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What is the relationship between zeros and factors of a polynomial?

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If x = r is a zero of the polynomial f(x), then (x - r) is a factor of f(x).

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How do you determine the degree of a polynomial?

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The degree of a polynomial is determined by the highest power of the variable in the polynomial expression.

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

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The leading coefficient (the coefficient of the x^3 term) determines the end behavior of the cubic function's graph.

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What does it mean for (x + a) to be a factor of f(x)?

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If (x + a) is a factor of f(x), then x = -a is a zero of f(x), meaning f(-a) = 0.

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What is the Remainder Theorem?

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The Remainder Theorem states that if a polynomial f(x) is divided by (x - r), the remainder is f(r).

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What is the Factor Theorem?

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The Factor Theorem states that (x - r) is a factor of f(x) if and only if f(r) = 0.

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How can you find the zeros of a cubic function?

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To find the zeros of a cubic function, you can use factoring, synthetic division, or numerical methods such as the Newton-Raphson method.

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What is the general shape of the graph of a cubic function?

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The graph of a cubic function can have one or two turning points and can extend to positive or negative infinity in both directions.