Solve frac{(x-sqrt{2})(x+sqrt{2})^2}{x^2-2}

$$\frac { ( x - \sqrt { 2 } ) ( x + \sqrt { 2 } ) ^ { 2 } } { x ^ { 2 } - 2 }$$

Evaluate

$$\frac{\left(x-\sqrt{2}\right)\left(x+\sqrt{2}\right)^{2}}{x^{2}-2}$$

Short Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+\sqrt{2}\right)^{2}.

$$\frac{\left(x-\sqrt{2}\right)\left(x^{2}+2x\sqrt{2}+\left(\sqrt{2}\right)^{2}\right)}{x^{2}-2}$$

The square of \sqrt{2} is 2.

$$\frac{\left(x-\sqrt{2}\right)\left(x^{2}+2x\sqrt{2}+2\right)}{x^{2}-2}$$

Use the distributive property to multiply x-\sqrt{2} by x^{2}+2x\sqrt{2}+2 and combine like terms.

$$\frac{x^{3}+\sqrt{2}x^{2}+2x-2x\left(\sqrt{2}\right)^{2}-2\sqrt{2}}{x^{2}-2}$$

The square of \sqrt{2} is 2.

$$\frac{x^{3}+\sqrt{2}x^{2}+2x-2x\times 2-2\sqrt{2}}{x^{2}-2}$$

Multiply -2 and 2 to get -4.

$$\frac{x^{3}+\sqrt{2}x^{2}+2x-4x-2\sqrt{2}}{x^{2}-2}$$

Combine 2x and -4x to get -2x.

$$\frac{x^{3}+\sqrt{2}x^{2}-2x-2\sqrt{2}}{x^{2}-2}$$

Differentiate w.r.t. x

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