Answer:
To factor and solve for the zeroes of the polynomial \(5x^4 + 5x^3 - 33x^2 - 3x + 18\), we can use the rational root theorem to identify potential rational roots. Then, we can use synthetic division or polynomial long division to divide the polynomial by these potential roots until we find the actual zeroes.
The rational root theorem states that if a polynomial equation has a rational root \( \frac{p}{q} \), where \( p \) is a factor of the constant term and \( q \) is a factor of the leading coefficient, then \( p \) is a factor of the constant term and \( q \) is a factor of the leading coefficient.
The constant term of the polynomial is 18, and the leading coefficient is 5. So, the potential rational roots are the factors of 18 divided by the factors of 5, which are:
\[ \pm 1, \pm 2, \pm 3, \pm 6, \pm 9, \pm 18 \]
Now, we can use synthetic division to test these potential roots. We'll start with \( x = 1 \):
\[
\begin{array}{c|ccccc}
1 & 5 & 5 & -33 & -3 & 18 \\
\hline
& & 5 & 10 & -23 & -26 \\
\end{array}
\]
Since the remainder is not zero, \( x = 1 \) is not a root. Let's try \( x = -1 \):
\[
\begin{array}{c|ccccc}
-1 & 5 & 5 & -33 & -3 & 18 \\
\hline
& & 5 & 0 & 33 & 36 \\
\end{array}
\]
Again, the remainder is not zero, so \( x = -1 \) is not a root.
We continue this process until we find a root that gives us a remainder of zero. Once we find a root, we can use polynomial division or synthetic division to divide the polynomial by the root to find the other roots.
