Polynomial and Rational Functions Review

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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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What determines the end behavior of a polynomial function?

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The end behavior of a polynomial function is determined by the degree of the polynomial and the sign of the leading coefficient.

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What is the Remainder Theorem?

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

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How do you find the possible rational zeros of a polynomial?

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To find the possible rational zeros of a polynomial, use the Rational Root Theorem, which states that any rational solution, expressed as p/q, has p as a factor of the constant term and q as a factor of the leading coefficient.

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What is an asymptote?

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An asymptote is a line that a graph approaches but never touches or crosses. It can be vertical, horizontal, or oblique.

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True or False: A graph can cross a vertical asymptote.

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False. A graph cannot cross a vertical asymptote.

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

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The leading coefficient determines the direction of the graph's end behavior. If it is positive, the graph rises to the right; if negative, it falls to the right.

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What does it mean for a polynomial to have an even degree?

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A polynomial has an even degree if the highest exponent of its variable is an even number. This typically results in the graph having the same end behavior on both sides.

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What does it mean for a polynomial to have an odd degree?

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A polynomial has an odd degree if the highest exponent of its variable is an odd number. This typically results in the graph having opposite end behaviors on both sides.

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How can you determine the number of positive real solutions of a polynomial?

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You can use Descartes' Rule of Signs, which states that the number of positive real roots is equal to the number of sign changes in f(x) or less than that by an even number.