Descartes, and Factoring Polynomials

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What is Descartes' Rule of Signs?

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

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How do you determine the number of positive roots using Descartes' Rule of Signs?

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Count the number of sign changes in the polynomial's coefficients. The number of positive roots is equal to the number of sign changes or less than that by an even number.

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How do you determine the number of negative roots using Descartes' Rule of Signs?

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To find the number of negative roots, substitute x with -x in the polynomial and count the sign changes in the resulting polynomial.

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What is synthetic division?

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Synthetic division is a simplified form of polynomial long division that is used to divide a polynomial by a linear factor of the form (x - c).

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What does it mean if a division problem is incorrect in polynomial division?

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It means that the quotient or remainder does not accurately represent the result of dividing the polynomial by the given divisor.

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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. If one zero is known, use polynomial division to factor the polynomial.

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Given one zero of a polynomial, how can you find the other zeros?

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Use synthetic division or polynomial long division to divide the polynomial by the factor corresponding to the known zero, then solve the resulting polynomial for its zeros.

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What is the significance of the degree of a polynomial in relation to its roots?

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The degree of a polynomial indicates the maximum number of roots (real and complex) it can have. A polynomial of degree n can have up to n roots.

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

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The number of positive roots can be equal to the number of sign changes in the polynomial or less than that by an even number.