Let y=x8+x4+1x4+x2+1.If dydx=ax3+bx. Then

y=x8+x4+1x4+x2+1.=x8+2x4+1-x4x4+x2+1=x4+12-x22x4+x2+1y=x4+x2+1x4-x2+1x4+x2+1 ⇒y=x4−x2+1dydx=4x3-2xa=4,b=-2

Let y=x8+x4+1x4+x2+1.If dydx=ax3+bx. Then

y=x8+x4+1x4+x2+1.=x8+2x4+1-x4x4+x2+1=x4+12-x22x4+x2+1y=x4+x2+1x4-x2+1x4+x2+1 ⇒y=x4−x2+1dydx=4x3-2xa=4,b=-2

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Let $y = \frac{x^{8} + x^{4} + 1}{x^{4} + x^{2} + 1} .$ If $\frac{d y}{d x} = a x^{3} + b x .$ Then

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$a = 4, b = 2$

$a = 4, b = - 2$

$a = - 2, b = 4$

$a = 2, b = - 4$

answer is B.

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Detailed Solution

$y = \frac{x^{8} + x^{4} + 1}{x^{4} + x^{2} + 1} . = \frac{(x^{8} + 2 x^{4} + 1) - x^{4}}{x^{4} + x^{2} + 1} = \frac{{(x^{4} + 1)}^{2} - {(x^{2})}^{2}}{x^{4} + x^{2} + 1}$

$y = \frac{(x^{4} + x^{2} + 1) (x^{4} - x^{2} + 1)}{x^{4} + x^{2} + 1}$

$\Rightarrow y = x^{4} - x^{2} + 1$

$\frac{d y}{d x} = 4 x^{3} - 2 x$

$a = 4, b = - 2$

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Let y=x8+x4+1x4+x2+1.If dydx=ax3+bx. Then

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Let $y = \frac{x^{8} + x^{4} + 1}{x^{4} + x^{2} + 1} .$ If $\frac{d y}{d x} = a x^{3} + b x .$ Then