Determinant

Elena09 Mar Ian 15, 2013 10:50 pm

1. Se considera functia f:R->R, $f(x)=\begin{vmatrix} x^3 &x^2 &x &1 \\ 1& 2& -1& 1\\ -1& 1& 5& 3\\ 4& 1& 0 & 0 \end{vmatrix}$

a) Sa se rezolve ecuatia f(x)=0.
b) Sa se discute numarul solutiilor reale ale ecuatiei f(x)=8m(x^2-1), m apartine R.

**Admin** Mar Ian 15, 2013 11:40 pm

Sub-punctul a) - rezolvare completa:

$\begin{array}{l} d = \left| {\begin{array}{*{20}{c}} {{x^3}}&{{x^2}}&x&1\\ 1&2&{ - 1}&1\\ { - 1}&1&5&3\\ 4&1&0&0 \end{array}} \right| = {x^3}\left| {\begin{array}{*{20}{c}} 2&{ - 1}&1\\ 1&5&3\\ 1&0&0 \end{array}} \right| + \left( { - 1} \right){x^2}\left| {\begin{array}{*{20}{c}} 1&{ - 1}&1\\ { - 1}&5&3\\ 4&0&0 \end{array}} \right|\\ \\ + x\left| {\begin{array}{*{20}{c}} 1&2&1\\ { - 1}&1&3\\ 4&1&0 \end{array}} \right| + \left( { - 1} \right)\left| {\begin{array}{*{20}{c}} 1&2&{ - 1}\\ { - 1}&1&5\\ 4&1&0 \end{array}} \right| \end{array}$

$\begin{array}{l} \left| {\begin{array}{*{20}{c}} 2&{ - 1}&1\\ 1&5&3\\ 1&0&0 \end{array}} \right| = - 8\\ \\ \left| {\begin{array}{*{20}{c}} 1&{ - 1}&1\\ { - 1}&5&3\\ 4&0&0 \end{array}} \right| = - 32\\ \\ \left| {\begin{array}{*{20}{c}} 1&2&1\\ { - 1}&1&3\\ 4&1&0 \end{array}} \right| = 16\\ \\ \left| {\begin{array}{*{20}{c}} 1&2&{ - 1}\\ { - 1}&1&5\\ 4&1&0 \end{array}} \right| = 40 \end{array}$

$\Rightarrow d = - 8{x^3} + 32{x^2} + 16x - 40 = 0 \Leftrightarrow - 8\left( {x - 1} \right)\left( {{x^2} - 3x - 5} \right) = 0 \Rightarrow {x_{1,2,3}} = ...$

Sub-punctul b) Tema !

Determinant

Determinant

Re: Determinant