Polynomial Zeros

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What is a polynomial zero?

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A polynomial zero is a value of x that makes the polynomial equal to zero. It is also known as a root of the polynomial.

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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 has at least one complex root.

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What does it mean for a zero to have multiplicity?

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

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How do you find the zeros of a polynomial?

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To find the zeros of a polynomial, set the polynomial equal to zero and solve for x. This can involve factoring, using the quadratic formula, or other algebraic methods.

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What is the end behavior of a polynomial function?

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The end behavior of a polynomial function describes how the function behaves as x approaches positive or negative infinity. It is determined by the leading term of the polynomial.

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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 influences the end behavior of the polynomial.

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What is the difference between rational and irrational zeros?

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Rational zeros can be expressed as a fraction of two integers, while irrational zeros cannot be expressed as a simple fraction and often involve square roots.

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What is a complex zero?

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A complex zero is a solution to a polynomial equation that includes imaginary numbers. Complex zeros always occur in conjugate pairs.

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

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

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What does it mean for zeros to come in pairs?

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Zeros that involve square roots or complex numbers often come in pairs due to the nature of polynomial equations and their factors.