Flashcard 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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How do you determine the degree of a polynomial?

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

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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 f(x) = 2x^3 - 3x^2 + 4, the leading coefficient is 2.

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What are x-intercepts of a polynomial function?

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X-intercepts are the points where the graph of the polynomial crosses the x-axis. They occur where f(x) = 0.

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How can you find the x-intercepts of a polynomial function?

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To find the x-intercepts, set the polynomial equal to zero and solve for x. For example, for f(x) = x^2 - 4, set x^2 - 4 = 0.

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What is the maximum number of x-intercepts a polynomial can have?

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A polynomial of degree n can have at most n x-intercepts.

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

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The leading term determines the end behavior of the polynomial function. For example, if the leading term is positive and of even degree, the ends of the graph will rise.

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What is the relationship between the degree of a polynomial and the number of turns in its graph?

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The maximum number of turns in the graph of a polynomial function is one less than its degree. For a cubic polynomial (degree 3), the maximum number of turns is 2.

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How do you use the leading-term test to determine the end behavior of a polynomial?

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The leading-term test involves analyzing the leading term's degree and coefficient. If the leading term is positive and of even degree, the graph rises on both ends.

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What is synthetic division and when is it used?

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Synthetic division is a simplified form of polynomial long division used to divide a polynomial by a linear factor of the form (x - c). It is faster and easier than long division.