1 problem found
Let \[ \f(x) = x^n + a_1 x^{n-1} + \cdots + a_n\;, \] where \(a_1\), \(a_2\), \(\ldots\), \(a_n\) are given numbers. It is given that \(\f(x)\) can be written in the form \[ \f(x) = (x+k_1)(x+k_2)\cdots(x+k_n)\;. \] By considering \(\f(0)\), or otherwise, show that \(k_1k_2 \ldots k_n =a_n\). 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 \((k_1-1)(k_2-1)\cdots(k_n-1)\). Find the roots of the equation \[ x^4 +22x^3 +172x^2 +552x+576=0\;, \] given that they are all integers.
Solution: \begin{align*} && f(0) &= 0^n + a_1\cdot 0^{n-1} + \cdots + a_n \\ &&&= a_n \\ && f(0) &= (0+k_1)(0+k_2) \cdots (0+k_n) \\ &&&= k_1 k_2 \cdots k_n \\ \Rightarrow && a_n &= k_1 k_2 \cdots k_n \\ \\ &&f(1) &= 1^n + a_1 \cdot 1^{n-1} + \cdots + a_n \\ &&&= 1 + a_1 + a_2 + \cdots + a_n \\ && f(1) &= (1 + k_1) (1 + k_2) \cdots (1+k_n) \\ \Rightarrow && (k_1+1)\cdots (k_n+1) &= 1 + a_1 + \cdots + a_n \\ \\ && f(-1) &= (-1)^{n} + a_1 \cdot (-1)^{n-1} + \cdots + a_n \\ &&&= a_n - a_{n-1} + \cdots + (-1)^{n-1} a_1 + (-1)^{n} \\ && f(-1) &= (-1+k_1)(-1+k_2) \cdots (-1+k_n) \\ &&&= (k_1-1)(k_2-1)\cdots(k_n-1) \\ \Rightarrow && (k_1-1)\cdots(k_n-1) &= a_n - a_{n-1} + \cdots + (-1)^{n-1} a_1 + (-1)^{n} \end{align*} \(576 = 2^6 \cdot 3^2\). Notice that \(1 - 22 + 172 -552 + 576 = 175 = 5^2 \cdot 7\) and \(1+22 + 172+552+576 = 1323 = 3^3 \cdot 7^2\). \(k_i = 2, 6, 6, 8\) therefore the roots are \(-2, -6, -6, -8\)