Question

decomponha as frações seguintes numa adição de frações, com denominadores do 1° grau. 1) 1/ (x-1)(x-2)(x-3)

Answer 1

$\dfrac{1}{(x-1)(x-2)(x-3)}=\dfrac{A}{x-1}+\dfrac{B}{x-2}+\dfrac{C}{x-3}\\ \\ \\ \dfrac{1}{(x-1)(x-2)(x-3)}=\cdots\\ \\ \hspace*{1cm}\cdots=\dfrac{A(x-2)(x-3)+B(x-1)(x-3)+C(x-1)(x-2)}{(x-1)(x-2)(x-3)}\\ \\ \\ (A+B+C)x^2-(5A+4B+3C)x+(6A+3B+2C)=1$


$\begin{cases} A+B+C=0\\ 5A+4B+3C=0\\ 6A+3B+2C=1 \end{cases}\\ \\ \\ a=\dfrac{1}{2}~,~b=-1~,~c=\dfrac{1}{2}\\ \\ \\ \dfrac{1}{(x-1)(x-2)(x-3)}=\dfrac{\frac{1}{2}}{x-1}-\dfrac{1}{x-2}+\dfrac{\frac{1}{2}}{x-3}\\ \\ \\ \boxed{\boxed{\dfrac{1}{(x-1)(x-2)(x-3)}=\dfrac{1}{2(x-1)}-\dfrac{1}{x-2}+\dfrac{1}{2(x-3)}}}$