Question

•Determina os valores de A, B e C para se observar em cada caso (o exercício é esse da foto anexada)

Gente me ajudem com esse exercício por favor, preciso para hoje, conto com a vossa ajuda.​

Answer 1

Continuação das alternativas

d)  $\frac{1}{(2n-1)(2n+1)} = \frac{A}{2n-1} + \frac{B}{2n+1}$

$Para\;A:\\\\\frac{A}{2n-1} = \frac{1}{(2n-1)(2n+1)} - \frac{B}{2n+1}\\\\A = (\frac{1}{(2n-1)(2n+1)} - \frac{B}{2n+1}) \cdot (2n-1)\\\\A = \frac{1\cdot (2n-1)}{(2n-1)(2n+1)} - \frac{B\cdot (2n-1)}{2n+1}\\\\A = \frac{1}{(2n+1)} - \frac{B(2n-1)}{2n+1}\\\\A = \frac{1-B(2n-1)}{(2n+1)}; \;\; para\;x \neq \frac{1}{2}$       $Para\;B:\\\\\frac{B}{2n+1} = \frac{1}{(2n-1)(2n+1)} - \frac{A}{2n-1}\\\\B = (\frac{1}{(2n-1)(2n+1)} - \frac{A}{2n-1}) \cdot (2n+1)\\\\B = \frac{1 \cdot (2n+1)}{(2n-1)(2n+1)} - \frac{A \cdot (2n+1)}{2n-1}\\\\B = \frac{1}{(2n-1)} - \frac{A(2n+1)}{2n-1}\\\\B = \frac{1-A(2n+1)}{(2n-1)}; \;\; para\;x \neq -\frac{1}{2}$

e)  $\frac{6-7x}{x^3-5x^2+6x} = \frac{A}{x} + \frac{B}{x-2} + \frac{C}{x-3}$

$Para\; A:\\\\\frac{6-7x}{(x)(x-2)(x-3)} = \frac{A}{x} + \frac{B}{x-2} + \frac{C}{x-3}\\\\\frac{A}{x} = \frac{6-7x}{(x)(x-2)(x-3)} - \frac{B}{x-2} - \frac{C}{x-3}\\\\A = (\frac{6-7x}{(x)(x-2)(x-3)} - \frac{B}{x-2} - \frac{C}{x-3}) \cdot (x)\\\\A = \frac{(6-7x)\cdot (x)}{(x)(x-2)(x-3)} - \frac{B\cdot (x)}{x-2} - \frac{C\cdot (x)}{x-3}\\\\A = \frac{6-7x}{(x-2)(x-3)} - \frac{B(x)}{x-2} - \frac{C(x)}{x-3}\\\\A = \frac{6-7x}{(x-2)(x-3)} - \frac{B(x)(x-3) - C(x)(x-2)}{(x-2)(x-3)}$

$A = \frac{6-7x-B(x^2-3x) - C(x^2-2x)}{(x-2)(x-3)}; \;\; para\; x \neq 0$

$Para\; B:\\\\\frac{6-7x}{(x)(x-2)(x-3)} = \frac{A}{x} + \frac{B}{x-2} + \frac{C}{x-3}\\\\\frac{B}{x-2} = \frac{6-7x}{(x)(x-2)(x-3)} - \frac{A}{x} - \frac{C}{x-3}\\\\B = (\frac{6-7x}{(x)(x-2)(x-3)} - \frac{A}{x} - \frac{C}{x-3}) \cdot (x-2)\\\\B = \frac{(6-7x)\cdot (x-2)}{(x)(x-2)(x-3)} - \frac{A\cdot (x-2)}{x} - \frac{C\cdot (x-2)}{x-3}\\\\B = \frac{6-7x}{(x)(x-3)} - \frac{A(x-2)}{x} - \frac{C(x-2)}{x-3}\\\\B = \frac{6-7x}{(x)(x-3)} - \frac{A(x-2)(x-3)-C(x-2)(x)}{(x)(x-3)}$

$B = \frac{6-7x - A(x-2)(x-3)-C(x-2)(x)}{(x)(x-3)}; \;\; para\; x \neq 2$

$Para\; C:\\\\\frac{6-7x}{(x)(x-2)(x-3)} = \frac{A}{x} + \frac{B}{x-2} + \frac{C}{x-3}\\\\\frac{C}{x-3} = \frac{6-7x}{(x)(x-2)(x-3)} - \frac{A}{x} - \frac{B}{x-2}\\\\C = (\frac{6-7x}{(x)(x-2)(x-3)} - \frac{A}{x} - \frac{B}{x-2}) \cdot (x-3)\\\\C = \frac{(6-7x)\cdot (x-3)}{(x)(x-2)(x-3)} - \frac{A\cdot (x-3)}{x} - \frac{B\cdot (x-3)}{x-2}\\\\C = \frac{6-7x}{(x)(x-2)} - \frac{A(x-3)}{x} - \frac{B(x-3)}{x-2}\\\\C = \frac{6-7x}{(x)(x-2)} - \frac{A(x-3)(x-2)-B(x-3)(x)}{(x)(x-2)}$

$C = \frac{6-7x - A(x-3)(x-2)-B(x-3)(x)}{(x)(x-2)}; \;\; para\; x \neq 3$