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) = 4x^3 + 2x^2 - x + 7, the degree is 3.

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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 the leading term (the term with the highest degree). If the leading coefficient is positive and the degree is even, the ends of the graph rise. If the leading coefficient is negative and the degree is even, the ends fall. If the degree is odd, one end rises and the other falls.

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What is the y-intercept of a polynomial function?

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The y-intercept of a polynomial function is the point where the graph intersects the y-axis. It can be found by evaluating the function at x = 0.

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What does it mean for a polynomial to be in standard form?

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A polynomial is in standard form when its terms are arranged in descending order of their degree. For example, f(x) = 2x^4 - 3x^3 + x - 5 is in standard form.

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

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The leading coefficient affects the end behavior of the polynomial function. It determines whether the graph rises or falls as x approaches positive or negative infinity.

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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 can you find the zeros of a polynomial function?

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Zeros can be found by factoring the polynomial, using the quadratic formula (for quadratics), or applying numerical methods for higher-degree polynomials.

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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 the difference between even and odd degree polynomials in terms of end behavior?

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