Zeros of Polynomial Functions

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What is the degree of a polynomial function?

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The degree of a polynomial function is the highest power of the variable in the polynomial. For example, in P(x) = 3x^4 - 7x^2 - 2x^7 - x + 4, the degree is 7.

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How do you determine the number of negative zeros in a polynomial function?

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To determine the number of negative zeros, use Descartes' Rule of Signs. Count the number of sign changes in P(-x). Each sign change indicates a possible negative zero.

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What are the zeros of a polynomial function?

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The zeros of a polynomial function are the values of x for which P(x) = 0. They are the x-intercepts of the graph of the function.

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How do you find the zeros of a polynomial expressed in factored form?

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To find the zeros of a polynomial in factored form, set each factor equal to zero and solve for x. For example, for f(x) = (x+4)(x-3)(x-2), the zeros are x = -4, 3, 2.

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What is the significance of complex zeros in polynomial functions?

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Complex zeros occur in conjugate pairs and indicate that the polynomial does not intersect the x-axis at those points. For example, if a polynomial has zeros -6, 3i, and -3i, it has both real and complex zeros.

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What is Descartes' Rule of Signs?

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Descartes' Rule of Signs is a method to determine the possible number of positive and negative real zeros of a polynomial by counting the number of sign changes in the polynomial's coefficients.

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How do you find the real zeros of a cubic polynomial?

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To find the real zeros of a cubic polynomial, you can use synthetic division, the Rational Root Theorem, or numerical methods like the Newton-Raphson method.

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What is the Rational Root Theorem?

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The Rational Root Theorem states that any rational solution (or root) of a polynomial equation is of the form ±p/q, where p is a factor of the constant term and q is a factor of the leading coefficient.

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What is a polynomial function?

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A polynomial function is a function that can be expressed in the form P(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + 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 relationship between the degree of a polynomial and the number of zeros?

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The Fundamental Theorem of Algebra states that a polynomial of degree n has exactly n zeros, counting multiplicities and including complex zeros.