Solve x^3-3x/x+sqrt{3}

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

Evaluate

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

Short Solution Steps

Rationalize the denominator of \frac{x^{3}-3x}{x+\sqrt{3}} by multiplying numerator and denominator by x-\sqrt{3}.

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

Consider \left(x+\sqrt{3}\right)\left(x-\sqrt{3}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

The square of \sqrt{3} is 3.

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

Factor the expressions that are not already factored.

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

Cancel out x^{2}-3 in both numerator and denominator.

$$x\left(x-\sqrt{3}\right)$$

Expand the expression.

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

Factor

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

Solution Steps

Rationalize the denominator of \frac{x^{3}-3x}{x+\sqrt{3}} by multiplying numerator and denominator by x-\sqrt{3}.

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

Consider \left(x+\sqrt{3}\right)\left(x-\sqrt{3}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

The square of \sqrt{3} is 3.

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

Factor the expressions that are not already factored in \frac{\left(x^{3}-3x\right)\left(x-\sqrt{3}\right)}{x^{2}-3}.

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

Cancel out x^{2}-3 in both numerator and denominator.

$$factor(x\left(x-\sqrt{3}\right))$$

Expand the expression.

$$factor(x^{2}-\sqrt{3}x)$$

Factor out x.

$$x\left(x-\sqrt{3}\right)$$