multiplicity and polynomials

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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 n is a non-negative integer.

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What are the zeros of a polynomial?

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The zeros of a polynomial are the values of x for which the polynomial evaluates to zero. They are also known as the roots of the polynomial.

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What does 'multiplicity' mean in relation to polynomial zeros?

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Multiplicity refers to the number of times a particular zero is repeated in a polynomial. For example, if (x - 3) appears twice in the factored form, then 3 is a zero with multiplicity 2.

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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 f(x) = 2x^3 + 3x^2 + 1, the degree is 3.

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

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The degree of a polynomial determines the maximum number of roots (or zeros) it can have. For example, a polynomial of degree 4 can have up to 4 roots.

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Write the polynomial function for zeros at 2 (multiplicity 3) and -1 (multiplicity 1).

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f(x) = (x - 2)^3 * (x + 1) or f(x) = (x + 1)(x - 2)^3.

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What is the factored form of a polynomial?

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The factored form of a polynomial expresses it as a product of its linear factors. For example, f(x) = (x - 1)(x - 2)(x - 3) is a factored form.

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How do you determine the multiplicity of a zero from a polynomial's factored form?

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The multiplicity of a zero can be determined by the exponent of its corresponding factor in the factored form. For example, in (x - 3)^2, the zero 3 has a multiplicity of 2.

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What is the significance of a zero with an even multiplicity?

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A zero with an even multiplicity touches the x-axis at that point but does not cross it, indicating a local minimum or maximum.

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What is the significance of a zero with an odd multiplicity?

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A zero with an odd multiplicity crosses the x-axis at that point, indicating a change in sign of the polynomial.