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_n*x^n + a_(n-1)*x^(n-1) + ... + a_1*x + a_0, where 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 the polynomial 4x^3 + 2x^2 - x + 7, the degree is 3.

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What does it mean for (x - c) to be a factor of a polynomial?

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If (x - c) is a factor of a polynomial P(x), then P(c) = 0. This means that c is a root of the polynomial.

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How do you perform synthetic division?

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Synthetic division is a shortcut method for dividing a polynomial by a linear factor of the form (x - c). It involves using the coefficients of the polynomial and performing a series of multiplications and additions.

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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 roots of a polynomial?

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To find the roots 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 the terms, such as 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 is the coefficient of the term with the highest degree in a polynomial. It determines the end behavior of the polynomial function.