Solving Polynomials

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What is the multiplicity of a root in a polynomial?

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The multiplicity of a root refers to the number of times a particular root appears in the factorization of a polynomial. For example, in the polynomial (x+1)², the root x=-1 has a multiplicity of 2.

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How do you find the zeros of a polynomial function?

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Zeros of a polynomial function can be found by factoring the polynomial, using the quadratic formula, or graphing the function to identify x-intercepts.

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What does it mean if a polynomial has a root with an even multiplicity?

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If a polynomial has a root with an even multiplicity, the graph of the polynomial will touch the x-axis at that root but will not cross it.

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What does it mean if a polynomial has a root with an odd multiplicity?

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If a polynomial has a root with an odd multiplicity, the graph of the polynomial will cross the x-axis at that root.

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What is the Fundamental Theorem of Algebra?

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The Fundamental Theorem of Algebra states that every non-constant polynomial function of degree n has exactly n roots in the complex number system, counting multiplicities.

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How can you determine the number of real zeros of a polynomial?

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The number of real zeros can be determined by analyzing the degree of the polynomial and using Descartes' Rule of Signs, along with graphing the polynomial.

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What is the quadratic formula?

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What is the significance of the leading coefficient in a polynomial?

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The leading coefficient determines the end behavior of the polynomial function. If it is positive, the graph rises to the right; if negative, it falls to the right.

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What is synthetic division and when is it used?

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Synthetic division is a simplified form of polynomial long division used to divide a polynomial by a linear factor of the form (x - c). It is quicker and easier than long division.

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What is a polynomial's degree?

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The degree of a polynomial is the highest power of the variable in the polynomial. It indicates the maximum number of roots the polynomial can have.