POLYNOMIAL FUNCTIONS

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What is a polynomial function?

Back

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.

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What are the degrees of polynomial functions?

Back

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 end behavior of polynomial functions?

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The end behavior of a polynomial function describes how the function behaves as x approaches positive or negative infinity. It is determined by the degree and leading coefficient.

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What is a relative extremum?

Back

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

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

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To find relative extrema, take the derivative of the polynomial, set it to zero to find critical points, and use the second derivative test or first derivative test to determine if they are 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.

Front

What is the leading coefficient of a polynomial?

Back

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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What is the difference between even and odd degree polynomials?

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Even degree polynomials have symmetric end behavior (both ends go up or down), while odd degree polynomials have opposite end behavior (one end goes up and the other goes down).

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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 the terms, such as f(x) = a_n*x^n + a_(n-1)*x^(n-1) + ... + a_1*x + a_0.

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How do you convert from factored form to standard form?

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To convert from factored form to standard form, expand the product of the factors using the distributive property or the FOIL method.