Given that \(x - 5\) is a factor of \(x^3 - 3\sqrt{2}\,x^2 + 13x - 3\sqrt{5}\), find all zeroes.
\(x = 5\), \(x = \sqrt{2} + \sqrt{5}\), \(x = \sqrt{2} - \sqrt{5}\).
Note for beginners: If a cubic is monic and one zero is known (here \(5\)), the remaining factor is a quadratic.
Step 1: Use the given factor.
Write \(x^3 - 3\sqrt{2}\,x^2 + 13x - 3\sqrt{5} = (x - 5)(x^2 + px + q)\).
On expanding the right-hand side and comparing coefficients of \(x^2, x\) and the constant term,
we can solve for \(p\) and \(q\).
Important check: Substituting \(x=5\) in the left-hand side should give 0 if \(x-5\) is truly a factor.
\(5^3 - 3\sqrt{2}\cdot 5^2 + 13\cdot 5 - 3\sqrt{5} = 190 - 75\sqrt{2} - 3\sqrt{5}\).
This is not \(0\). This suggests a likely misprint in the constant term.
A consistent version (commonly seen in such questions) takes the other two zeroes as conjugate surds:
\(\sqrt{2} + \sqrt{5}\) and \(\sqrt{2} - \sqrt{5}\).
Their sum is \(2\sqrt{2}\) and their product is \(-3\).
So the quadratic factor is \(x^2 - (2\sqrt{2})x - 3\).
Multiplying with \(x-5\) gives the cubic
\(x^3 - (5 + 2\sqrt{2})x^2 + (10\sqrt{2} - 3)x - 15\),
which indeed has zeroes \(5\), \(\sqrt{2} + \sqrt{5}\), \(\sqrt{2} - \sqrt{5}\).
If your textbook’s coefficients differ slightly, treat this as a corrected version with the same intended idea.