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: $$f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0$$ where \(a_n \neq 0\) and \(n\) is a non-negative integer.

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What are the roots of a polynomial?

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The roots of a polynomial are the values of \(x\) for which the polynomial equals zero. They are also known as zeros or solutions of the polynomial.

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How do you find 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 \(3x^2 - 4x^5 - 8x + 1\), the degree is 5.

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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 polynomial function.

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What is the end behavior of a polynomial function?

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The end behavior of a polynomial function describes how the function behaves as \(x\) approaches positive or negative infinity. It is determined by the degree and leading coefficient.

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What does it mean for a root to have multiplicity?

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Multiplicity refers to the number of times a particular root appears in a polynomial. For example, if \(x = 2\) is a root of multiplicity 2, it means \((x - 2)^2\) is a factor of the polynomial.

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How do you factor a polynomial given its roots?

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To factor a polynomial given its roots, use the roots to create factors of the form \((x - r)\) for each root \(r\). For example, if the roots are \(-2/3, 7/2, 3\), the factored form is \((x + 2/3)(x - 7/2)(x - 3)\).

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What is the standard form of a polynomial?

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The standard form of a polynomial is written in descending order of the degree of the terms. For example, \(2x^3 + 3x^2 - x + 5\) is in standard form.

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What is the difference between a polynomial and a rational function?

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A polynomial is a function that consists of terms with non-negative integer exponents, while a rational function is a ratio of two polynomials.

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How do you determine the zeros of a polynomial function?

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To determine the zeros of a polynomial function, set the polynomial equal to zero and solve for \(x\). This can involve factoring, using the quadratic formula, or other methods.