[STEP][剑桥数学入学考试][02-S1-Q5][Roots of Polynomials][FP1] STEP

casperyc的马甲 2023-12-12 1264

Let f(x)=xn+a1xn1++an, \mathrm{f}(x) = x^n + a_1 x^{n-1} + \cdots + a_n, where a1,a2,,ana_1, a_2, \ldots, a_n are given numbers. It is given that f(x)\mathrm{f}(x) can be written in the form f(x)=(x+k1)(x+k2)(x+kn). \mathrm{f}(x) = (x+k_1)(x+k_2)\cdots(x+k_n).

  1. By considering f(0)\mathrm{f}(0), or otherwise, show that k1k2kn=ank_1k_2 \ldots k_n =a_n.
  2. Show also that (k1+1)(k2+1)(kn+1)=1+a1+a2++an (k_1+1)(k_2+1)\cdots(k_n+1)= 1+a_1+a_2+\cdots+a_n and give a corresponding result for (k11)(k21)(kn1)(k_1-1)(k_2-1)\cdots(k_n-1).
  3. Find the roots of the equation x4+22x3+172x2+552x+576=0, x^4 +22x^3 +172x^2 +552x+576=0, given that they are all integers.
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