Matrices & Determinants
Determinant Expansion
Grade None
Question:
<p>If \(\phi(x) = \begin{vmatrix} x^2 + 5x + 3 & 2x + 5 & 3 \\ 3x^2 - x - 4 & 6x - 1 & 9 \\ 7x^2 + 6x + 9 & 14x + 6 & 21 \end{vmatrix} = ax^3 + bx^2 + cx + d\), then</p>
<p>(a) \(a = 0\)</p>
<p>(b) \(b = 0\)</p>
<p>(c) \(c = 0\)</p>
<p>(d) \(d = 47\)</p>
Step-by-Step Solution
Key Concept: Recognize that the third column is a multiple of combinations of the first two columns, making this determinant vanish at high degree terms.
<p>Expand the determinant by factoring and observing that the first row operations show the determinant reduces to a polynomial. Analysis of the structure shows the coefficient of $x^3$ is zero.</p>
Correct Answer: A