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 a_n, a_(n-1), ..., a_0 are constants and n is a non-negative integer.

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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 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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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 difference between a monomial, binomial, and trinomial?

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A monomial is a polynomial with one term, a binomial has two terms, and a trinomial has three terms. For example, 3x is a monomial, 3x + 2 is a binomial, and 3x^2 + 2x + 1 is a trinomial.

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How do you add polynomials?

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To add polynomials, combine like terms by adding their coefficients. For example, (2x^2 + 3x) + (4x^2 + 5) = 6x^2 + 3x + 5.

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How do you multiply polynomials?

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To multiply polynomials, use the distributive property (FOIL for binomials) to multiply each term in the first polynomial by each term in the second polynomial.

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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 involves using coefficients.