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

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What does it mean for a polynomial to be even?

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A polynomial is even if f(-x) = f(x) for all x in its domain. This means the graph of the polynomial is symmetric with respect to the y-axis.

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What does it mean for a polynomial to be odd?

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A polynomial is odd if f(-x) = -f(x) for all x in its domain. This means the graph of the polynomial is symmetric with respect to the origin.

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How do you evaluate a polynomial function at a specific value?

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To evaluate a polynomial function at a specific value, substitute the value into the polynomial expression and simplify. For example, for f(x) = 3x^2 - 9x - 20, to find f(5), substitute 5 for x.

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What is a leading coefficient?

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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 polynomial's graph.

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What is the significance of the leading coefficient's sign?

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The sign of the leading coefficient affects the direction of the graph's ends: if positive, the graph rises to the right; if negative, it falls to the right.

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What is the difference between a monomial, binomial, and trinomial?

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A monomial has one term, a binomial has two terms, and a trinomial has three terms. For example, 3x^2 is a monomial, 3x^2 - 5 is a binomial, and 3x^2 - 5x + 2 is a trinomial.

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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 - c), the remainder of that division is f(c).

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What is the Factor Theorem?

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