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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.

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What does it mean for a function to be even?

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A function f(x) is even if for every x in the domain, f(-x) = f(x). This means the graph of the function is symmetric about the y-axis.

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What does it mean for a function to be odd?

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A function f(x) is odd if for every x in the domain, f(-x) = -f(x). This means the graph of the function is symmetric about the origin.

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How do you determine if a polynomial function is even, odd, or neither?

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To determine if a polynomial function is even, odd, or neither, substitute -x into the function and compare it to f(x) and -f(x). If it equals f(x), it's even; if it equals -f(x), it's odd; otherwise, it's neither.

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What is the multiplicity of a zero?

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The multiplicity of a zero refers to the number of times a particular zero (or root) appears in a polynomial function. A zero with an even multiplicity will touch the x-axis, while a zero with an odd multiplicity will cross the x-axis.

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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. It determines the end behavior of the graph.

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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. It indicates the maximum number of roots the polynomial can have.

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What is the Factor Theorem?

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The Factor Theorem states that if f(c) = 0 for a polynomial f(x), then (x - c) is a factor of f(x).

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What is synthetic division?

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Synthetic division is a simplified method of dividing a polynomial by a linear factor of the form (x - c). It is faster than long division and is used to find polynomial quotients and remainders.

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What is the Remainder Theorem?

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The Remainder Theorem states that the remainder of the division of a polynomial f(x) by (x - c) is equal to f(c).