Question

Determine the convenient x-values as defined in class for the following 3x-1 t? (х+3)(2x – 7) (x2+1) пит T + пит Т2 + num х+3 + num 2х7 num + 12+1 O7/2 0 3 -7/2 -3 2 7 -2 1 -1

Answer 1

We have,

$\small \frac{3x-1}{x^2\left(x+3\right)\left(2x-7\right)\left(x^2+1\right)}=\frac{a_0}{x}+\frac{a_1}{x^2}+\frac{a_2}{x+3}+\frac{a_3}{2x-7}+\frac{a_5x+a_4}{x^2+1}$

So, multiplying both sides by the denominator on the left, we get

$\small 3x-1\\ \\=a_0x\left(x+3\right)\left(2x-7\right)\left(x^2+1\right)+a_1\left(x+3\right)\left(2x-7\right)\left(x^2+1\right)\\ \\+a_2x^2\left(2x-7\right)\left(x^2+1\right)+a_3x^2\left(x+3\right)\left(x^2+1\right)\\ \\+x^2\left(a_5x+a_4\right)\left(x+3\right)\left(2x-7\right)$

Now, we choose values of x such that most of the expressions on the right become 0, so, the convenient choices for x are

$\small x=0,\;-3,\;{7\over 2}$