Question

DESCOMPONER EN SUS FRACCIONES PARCIALES

1) (- x ^ 3 + x ^ 2 + 7x - 3)/(x ^ 4 + 4x ^ 2 + 3)​

Answer 1

Respuesta:

$\frac{4x-2}{x^2+1}+\frac{-5x+3}{x^2+3}$

Explicación paso a paso:

$\frac{\left(-x^3+x^2+7x-3\right)}{\left(x^4+4x^2+3\right)}\\\\\frac{-x^3+x^2+7x-3}{\left(x^2+1\right)\left(x^2+3\right)}\\\\\frac{-x^3+x^2+7x-3}{\left(x^2+1\right)\left(x^2+3\right)}=\frac{a_1x+a_0}{x^2+1}+\frac{a_3x+a_2}{x^2+3}\\\\-x^3+x^2+7x-3=\left(a_1x+a_0\right)\left(x^2+3\right)+\left(a_3x+a_2\right)\left(x^2+1\right)$

$-x^3+x^2+7x-3=a_1x^3+3a_1x+a_0x^2+3a_0+a_3x^3+a_3x+a_2x^2+a_2\\\\-1\cdot \:x^3+1\cdot \:x^2+7x-3=x^3\left(a_1+a_3\right)+x^2\left(a_0+a_2\right)+x\left(3a_1+a_3\right)+\left(3a_0+a_2\right)$

$\begin{bmatrix}3a_0+a_2=-3\\ 3a_1+a_3=7\\ a_0+a_2=1\\ a_1+a_3=-1\end{bmatrix}\\\\a_0=-2,\:a_2=3,\:a_1=4,\:a_3=-5\\\\\frac{4x+\left(-2\right)}{x^2+1}+\frac{\left(-5\right)x+3}{x^2+3}\\\\\frac{4x-2}{x^2+1}+\frac{-5x+3}{x^2+3}$