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 and a_n ≠ 0.

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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 + 2x^3 - x + 5, the degree is 4.

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How many roots can a polynomial of degree n have?

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A polynomial of degree n can have at most n roots, counting multiplicities.

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What is synthetic division?

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Synthetic division is a simplified form of polynomial long division that is used to divide a polynomial by a linear factor of the form (x - c). It is faster and requires less writing.

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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.

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How do you find the zeros of a polynomial?

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To find the zeros of a polynomial, set the polynomial equal to zero and solve for the variable. This can involve factoring, using the quadratic formula, or synthetic division.

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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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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 its terms, such as P(x) = a_n*x^n + a_(n-1)*x^(n-1) + ... + a_1*x + a_0.

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What is the leading coefficient of a polynomial?

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The leading coefficient of a polynomial is the coefficient of the term with the highest degree. For example, in P(x) = 4x^3 + 2x^2 - x + 7, the leading coefficient is 4.