Factor theorem

Question:
If the polynomial, $P(x) = ax^4 + (b-a)x^3 + (1-2ab)x^2 + (ab-10)x + 2ab$, has a factor $x-2$, find $a$ and $b$.

Answer:
The factor theorem says, "If a polynomial, $P(x)$, has a factor, $x-\alpha$, then $P(\alpha) = 0$. In this problem, we can put $P(2) = 0$. Thus,
\begin{eqnarray*}
P(2) &=& 16a + 8(b-a) + 4(1-2ab) + (ab-10) + 2ab = 0   \\
&\rightarrow& 4ab - 8a - 8b + 16 = 0    \\
&\rightarrow& ab - 2a - 2b + 4 = 0    \\
&\rightarrow& a(b - 2) - 2(b - 2) = 0    \\
&\rightarrow& (a-2)(b-2) = 0    \\
\end{eqnarray*}
In order to have the factor $x-2$ for $P(x)$, $a=2$ and $b=2$.

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