[PDF] chinese remainder theorem for polynomials

Find a polynomial p(x) that simultaneously has both the following properties. (i) When p(x) is divided by x100 the remainder is the constant polynomial 1. (ii) When p(x) is divided by (x?2)3 the remainder is the constant polynomial 2.
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  • What is the remainder theorem for polynomial equations?

    This illustrates the Remainder Theorem. If a polynomial f(x) is divided by x?a , the remainder is the constant f(a) , and f(x)=q(x)?(x?a)+f(a) , where q(x) is a polynomial with degree one less than the degree of f(x) .
    Synthetic division is a simpler process for dividing a polynomial by a binomial.

  • What are the examples of using Chinese remainder theorem?

    Example: Solve the simultaneous congruences x ? 6 (mod 11), x ? 13 (mod 16), x ? 9 (mod 21), x ? 19 (mod 25).
    Solution: Since 11, 16, 21, and 25 are pairwise relatively prime, the Chinese Remainder Theorem tells us that there is a unique solution modulo m, where m = 11?16?21?25 = 92400.

  • What are the examples of using Chinese remainder theorem?

    The Chinese Remainder Theorem says that a ring element can be speci- fied by giving its congruence class modulo each ideal in a collection of pair- wise relatively prime ideals, and this element is unique modulo the product (intersection) of the ideals.

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