Hasil bagi polinomial \(3x^4 + (p - q)x^3 - 2x^2 + 2px + p - q + 1\) oleh \((x^2 + 4)\) memberikan sisa \((54 - 14x)\). Nilai dari \(p \:.\: q = \dotso\)
Jawab: E
Metode pembagian bersusun
\begin{array}{r}
3x^2 + (p - q)x - 14\\
x^2 + 4\enclose{longdiv}{3x^4 + (p - q)x^3 - 2x^2 + 2px + p - q + 1}\\
\underline{3x^4 + 12x^2}\hspace{12.5em}\\
(p - q)x^3 -14x^2 + 2px + p - q + 1\\
\underline{(p - q)x^3 + 4(p - q)x}\hspace{5.5em}\\
-14x^2 - (2p - 4q)x + p - q + 1\\
\underline{-14x^2 - 56}\hspace{8.3em}\\
- (2p - 4q)x + p - q + 57
\end{array}
Sisa hasil pembagian \(- (2p - 4q)x + p - q + 57 \equiv 54 - 14x\)
Sehingga \(- (2p - 4q) = -14\) dan \(p - q + 57 = 54\)
\begin{equation*}
\begin{split}
- (2p - 4q) & = -14 \\\\
p - 2q & = 7 \quad {\color {red} \dotso \: (1)}
\end{split}
\end{equation*}
\begin{equation*}
\begin{split}
p - q + 57 & = 54 \\\\
p - q & = -3 \quad {\color {red} \dotso \: (2)}
\end{split}
\end{equation*}
Eliminasi persamaan (1) dan (2)
\begin{equation*}
\begin{split}
p - 2q & = 7 \\\\
p - q & = -3 \quad (-) \\
\hline \\
-q & = 10 \\\\
q & = -10
\end{split}
\end{equation*}
Substitusi q = −10 ke persamaan (2)
\begin{equation*}
\begin{split}
p - q & = -3 \\\\
p + 10 & = -3 \\\\
p & = -13
\end{split}
\end{equation*}
