Matrices & Determinants
Determinants as Polynomials
Grade 12
Question:
<p>If one root of the equation <span style='display:inline-block; border: 1px solid black; padding: 5px;'>\[\begin{vmatrix} 7 & 6 & x^2-13 \\ 2 & x^2-13 & 2 \\ x^2-13 & 3 & 7 \end{vmatrix} = 0\]</span> is \(x = 2\), the sum of all other five roots is</p>
<p>(a) \(2\sqrt{15}\)</p>
<p>(b) \(-2\)</p>
<p>(c) \(\sqrt{20} + \sqrt{15} - 2\)</p>
<p>(d) None of these</p>
Step-by-Step Solution
Key Concept: The determinant is a polynomial in $x$. Use Vieta's formulas on the reduced polynomial after factoring out the known root.
<p>Expand the determinant to obtain a polynomial equation in $x$. Verify that $x = 2$ is a root. Factor out $(x-2)$ and use Vieta's formulas on the remaining quintic factor to find the sum of the other five roots.</p>
Correct Answer: B