Respuesta :
Answer:
To solve the polynomial equation \( x^2 - 8x^2 - 18x - 20 = 0 \), let's first use synthetic division to test for possible rational roots. The rational root theorem states that any rational root of the polynomial equation must be of the form \( \frac{p}{q} \), where \( p \) is a factor of the constant term (in this case, 20) and \( q \) is a factor of the leading coefficient (in this case, 1).
Let's try synthetic division with some potential rational roots:
1. \( x = 1 \):
\[
\begin{array}{c|cccc}
1 & 1 & -8 & -18 & -20 \\
\hline
& & 1 & -7 & -25 \\
\end{array}
\]
Since the remainder is not zero, \( x = 1 \) is not a root.
2. \( x = -1 \):
\[
\begin{array}{c|cccc}
-1 & 1 & -8 & -18 & -20 \\
\hline
& & -1 & 9 & 9 \\
\end{array}
\]
Since the remainder is not zero, \( x = -1 \) is not a root.
3. \( x = 2 \):
\[
\begin{array}{c|cccc}
2 & 1 & -8 & -18 & -20 \\
\hline
& & 2 & -28 & -64 \\
\end{array}
\]
Since the remainder is not zero, \( x = 2 \) is not a root.
4. \( x = -2 \):
\[
\begin{array}{c|cccc}
-2 & 1 & -8 & -18 & -20 \\
\hline
& & -2 & 4 & 8 \\
\end{array}
\]
Since the remainder is not zero, \( x = -2 \) is not a root.
After testing some potential rational roots, it seems none of them are roots of the polynomial equation. We might need to use other methods to find the roots of this equation. Would you like to proceed with another method or have any other instructions?
