Matrices & Determinants
Determinant Coefficients
Grade 12
Question:
<p>If \(\phi(x) = \begin{vmatrix} 4x + 4 & (x+2)^2 & x^3 \\ 8x + 4 & 2(x+2)^2 & (x+1)^3 \\ 12x + 4 & 3(x+2)^2 & (x+13) \end{vmatrix}\), then which statement about \(\phi(x)\) is correct?</p>
<p>(a) The term independent of \(x\) in \(\phi(x)\) is \(16(5 + \sqrt{2} + \sqrt{3})\)</p>
<p>(b) The coefficient of \(x\) in \(\phi(x)\) is \(48(1 + \sqrt{2} + \sqrt{3})\)</p>
<p>(c) The coefficient of \(x\) in \(\phi(x)\) is \(16(5 + \sqrt{2} + \sqrt{3})\)</p>
<p>(d) The coefficient of \(x\) in \(\phi(x)\) is divisible by \(16\)</p>
Step-by-Step Solution
Key Concept: Carefully expand the determinant and collect terms by powers of $x$ to find specific coefficients.
<p>Expand the determinant using cofactor expansion. Extract the coefficient of $x$ from the resulting polynomial and verify divisibility by $16$.</p>
Correct Answer: D