Inequalities, Polynomials, and 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 are the zeros of a polynomial function?

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The zeros of a polynomial function are the values of $$x$$ for which the function equals zero. They are also known as the roots of the polynomial.

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How do you find the zeros of a polynomial function?

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To find the zeros of a polynomial function, set the function equal to zero and solve for $$x$$. This may involve factoring, using the quadratic formula, or other algebraic methods.

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What is the relationship between the degree of a polynomial and its number of zeros?

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A polynomial of degree $$n$$ can have at most $$n$$ real zeros. If the polynomial has complex zeros, they occur in conjugate pairs.

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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 ratio of two polynomials: $$f(x) = \frac{p(x)}{q(x)}$$ where $$p(x)$$ and $$q(x)$$ are polynomials and $$q(x) eq 0$$.

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What is the process of simplifying a rational expression?

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To simplify a rational expression, factor both the numerator and the denominator and then cancel any common factors.

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What is function composition?

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Function composition is the process of applying one function to the results of another function. If $$f(x)$$ and $$g(x)$$ are two functions, the composition is denoted as $$f(g(x))$$.

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How do you evaluate a composite function?

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To evaluate a composite function $$f(g(x))$$, first find the value of $$g(x)$$ and then substitute that value into the function $$f$$.

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What is an inequality?

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An inequality is a mathematical statement that compares two expressions, showing that one is greater than, less than, greater than or equal to, or less than or equal to the other.

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What are the different types of inequalities?

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The main types of inequalities are: 1. Strict inequalities: $$a < b$$ or $$a > b$$ 2. Non-strict inequalities: $$a \leq b$$ or $$a \geq b$$.