Polynomial & Rational 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: $$f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0$$ where \(a_n, a_{n-1}, ..., a_0\) are constants and \(n\) 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 the polynomial $$2x^3 + 3x^2 - x + 5$$, the degree is 3.

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What is a rational function?

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A rational function is a function that can be expressed as the quotient of two polynomials. The general form is: $$f(x) = \frac{P(x)}{Q(x)}$$ where \(P(x)\) and \(Q(x)\) are polynomials and \(Q(x) \neq 0\).

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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 this division is equal to \(f(c)\).

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What is the Factor Theorem?

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

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How do you find the x-intercept of a function?

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To find the x-intercept of a function, set the function equal to zero and solve for \(x\). The x-intercept occurs where the graph of the function crosses the x-axis.

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What is a vertical asymptote?

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A vertical asymptote is a line \(x = a\) where a function approaches infinity or negative infinity as \(x\) approaches \(a\). It occurs when the denominator of a rational function is zero.

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What is a horizontal asymptote?

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A horizontal asymptote is a line \(y = b\) that the graph of a function approaches as \(x\) approaches infinity or negative infinity. It indicates the behavior of the function at extreme values of \(x\).

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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 root appears in a polynomial. For example, in the polynomial \((x - 2)^3\), the zero \(x = 2\) has a multiplicity of 3.

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What is a polynomial inequality?

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A polynomial inequality is an inequality that involves a polynomial expression. For example, \(f(x) > 0\) or \(f(x) \leq 0\) where \(f(x)\) is a polynomial.