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

casperyc的马甲 7月前 676

Let $$ \mathrm{f}(x) = x^n + a_1 x^{n-1} + \cdots + a_n, $$ where $a_1, a_2, \ldots, a_n$ are given numbers. It is given that $\mathrm{f}(x)$ can be written in the form $$ \mathrm{f}(x) = (x+k_1)(x+k_2)\cdots(x+k_n). $$

  1. By considering $\mathrm{f}(0)$, or otherwise, show that $k_1k_2 \ldots k_n =a_n$.
  2. Show also that $$ (k_1+1)(k_2+1)\cdots(k_n+1)= 1+a_1+a_2+\cdots+a_n $$ and give a corresponding result for $(k_1-1)(k_2-1)\cdots(k_n-1)$.
  3. Find the roots of the equation $$ x^4 +22x^3 +172x^2 +552x+576=0, $$ given that they are all integers.
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