Problem source

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 source

\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 $-2, -6, -6, -8$