POLYNOMIAL FUNCTIONS

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What is a polynomial function?

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A polynomial function is a mathematical expression that involves 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 f(x) = 2x^3 + 3x^2 + 1, the degree is 3.

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What is a leading coefficient?

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The leading coefficient is the coefficient of the term with the highest degree in a polynomial. For example, in f(x) = -4x^5 + 2x^3 + 1, the leading coefficient is -4.

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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. For example, f(x) = 3x^4 - 2x^3 + x - 5 is in standard form.

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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 be done using factoring, the quadratic formula, or numerical methods.

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What is factored form of a polynomial?

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Factored form expresses a polynomial as a product of its factors. For example, f(x) = (x - 1)(x + 2) is a factored form of a polynomial.

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What is the relationship between the degree and the number of roots?

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The degree of a polynomial indicates the maximum number of roots it can have. A polynomial of degree n can have up to n real roots.

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What is synthetic division?

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Synthetic division is a simplified method of dividing a polynomial by a linear divisor of the form (x - c). It is faster than long division and is used to find polynomial roots.

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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.