Question:

If the polynomial, \(P(x) = ax^4 + (b-a)x^3 + (1-2ab)x^2 + (ab-10)x + 2ab\), has a factor \(x-2\), then 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 plug into \(P(2) = 0\). Thus,

\[  P(2) = 16a + 8(b-a) + 4(1-2ab) + (ab-10) + 2ab = 0   \]

\[  \rightarrow 4ab – 8a – 8b + 16 = 0    \]

\[  \rightarrow ab – 2a – 2b + 4 = 0    \]

\[  \rightarrow a(b – 2) – 2(b – 2) = 0    \]

\[  \rightarrow (a-2)(b-2) = 0    \]

In order to have the factor of \(x-2\) for \(P(x)\), \(a=2\) and \(b=2\).