The polynomial
\[ \begin{align}
p(x) &= x^4 + x^3 - 11x^2 - 9x + 2k^2
\end{align} \ \]
(a) Find the value of k for which p(x) is divisible by both (x - 1) and (x - 3)
(b) Solve p(x) = 0 for this value of k.
The value of k must be a common solution to p(1) = 0 and p(3) = 0.
Solving p(1) = 0:
\[ \begin{align}
1 + 1 - 11 - 9 + 2k^2 &= 0 \\
2k^2 - 9 - 9 &= 0 \\
2k^2 &= 18
\end{align} \ \]
So k = -3 or +3.
Solving p(3) = 0:
\[ \begin{align}
81 + 27 - 99 - 27 + 2k^2 &= 0 \\
2k^2 - 18 &= 0 \\
k^2 &= 9
\end{align} \ \]
So k = -3 or +3 again.
Test the factors of the constant term to see if they satisfy p(x).
Setting k = 3,
\[ \begin{align}
p(-2) &= 16 - 8 - 44 + 18 + 18 \\
&= 0 \\
p(-3) &= 81 - 27 - 99 + 27 + 18 \\
&= 0
\end{align} \ \]
So (x+2) and (x+3) are factors, and
\[ \begin{align}
p(x) &= (x -1)(x + 2)(x - 3)(x + 3) \quad \quad \quad \quad
\end{align} \ \]