Evaluate the partial fraction decomposition o | vedclass

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(D) We start with the expression $\frac{2x}{x^4 + x^2 + 1}$.First,factor the denominator: $x^4 + x^2 + 1 = (x^4 + 2x^2 + 1) - x^2 = (x^2 + 1)^2 - x^2$

Vedclass · Accepted Answer

$\frac{1}{x^2 - x + 1} - \frac{1}{x^2 + x + 1}$

Vedclass · Answer

(D) We start with the expression $\frac{2x}{x^4 + x^2 + 1}$.First,factor the denominator: $x^4 + x^2 + 1 = (x^4 + 2x^2 + 1) - x^2 = (x^2 + 1)^2 - x^2$.Using the difference of squares formula $a^2 - b^2 = (a - b)(a + b)$,we get $(x^2 - x + 1)(x^2 + x + 1)$.Now,express the fraction as: $\frac{2x}{(x^2 - x + 1)(x^2 + x + 1)}$.Using partial fractions,we set $\frac{2x}{(x^2 - x + 1)(x^2 + x + 1)} = \frac{A}{x^2 - x + 1} + \frac{B}{x^2 + x + 1}$.By solving,we find the difference: $\frac{1}{x^2 - x + 1} - \frac{1}{x^2 + x + 1} = \frac{(x^2 + x + 1) - (x^2 - x + 1)}{(x^2 - x + 1)(x^2 + x + 1)} = \frac{2x}{x^4 + x^2 + 1}$.Thus,the correct option is $D$.

Evaluate the partial fraction decomposition of $\frac{2x}{x^4 + x^2 + 1}$.

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