Pré-Vestibular ⇒ (UFGD - 2012) Determinante

[fellipiortiz]

Dada a matriz A(x) = [tex3]\left(\begin{array}{cc} x & 1 & 4 & 6 \\ x & x & 2 & 5 \\ x & x & x & 3 \\ x & x & x & x \end{array}\right)[/tex3] considere P(x) = detA(x) em que "det" denota o determinante. Então o polinômio p(x) pode ser fatorado como

(A) x(x - 1)(x - 2)(x - 3)
(B) x(x - 2)(x - 3)(x - 4)
(C) x(x - 3)(x - 4)(x - 5)
(D) x(x - 4)(x - 5)(x - 6)
(E) x(x - 1)(x - 3)(x - 4)

Resposta

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(Resposta: A)

[Vinícius]

[tex3]\begin{vmatrix}x & 1 & 4 & 6 \\ x & x & 2 & 5 \\ x & x & x & 3 \\ x & x & x & x\end{vmatrix}[/tex3]

[tex3]\begin{vmatrix}x-1 & 1 & 4 & 6 \\ 0 & x & 2 & 5 \\ 0 & x & x & 3 \\ 0 & x & x & x\end{vmatrix}[/tex3]

[tex3]\begin{vmatrix}x-1 & x & 4 & 6 \\ 0 & x & 2 & 5 \\ 0 & x & x & 3 \\ 0 & x & x & x\end{vmatrix}[/tex3]

[tex3]x(x-1)\begin{vmatrix}1 & 1 & 4 & 6 \\ 0 & 1 & 2 & 5 \\ 0 & 1 & x & 3 \\ 0 & 1 & x & x\end{vmatrix}[/tex3]

[tex3]x(x-1)\begin{vmatrix}1 & 2 & 5 \\ 1 & x & 3 \\ 1 & x & x\end{vmatrix}[/tex3]

[tex3]x(x-1)\begin{vmatrix}x-2&-2\\x-2&x-5\end{vmatrix}[/tex3]

[tex3]x(x-1)(x-2)\begin{vmatrix}1&-2\\1&x-5\end{vmatrix}[/tex3]

[tex3]x(x-1)(x-2)(x-3)[/tex3]