Solve -x/x^6+3/x^2-1/x^6/1+frac{3{x^5}+frac{2}{{x}^2}}

$$\frac{ - \frac{ x }{ { x }^{ 6 } } + \frac{ 3 }{ { x }^{ 2 } } - \frac{ 1 }{ { x }^{ 6 } } }{ 1+ \frac{ 3 }{ { x }^{ 5 } } + \frac{ 2 }{ { x }^{ 2 } } }$$

Evaluate

$$\frac{3x^{4}-x-1}{x\left(x^{5}+2x^{3}+3\right)}$$

Short Solution Steps

Rewrite x^{6} as xx^{5}. Cancel out x in both numerator and denominator.

$$\frac{-\frac{1}{x^{5}}+\frac{3}{x^{2}}-\frac{1}{x^{6}}}{1+\frac{3}{x^{5}}+\frac{2}{x^{2}}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x^{5} and x^{2} is x^{5}. Multiply \frac{3}{x^{2}} times \frac{x^{3}}{x^{3}}.

$$\frac{-\frac{1}{x^{5}}+\frac{3x^{3}}{x^{5}}-\frac{1}{x^{6}}}{1+\frac{3}{x^{5}}+\frac{2}{x^{2}}}$$

Since -\frac{1}{x^{5}} and \frac{3x^{3}}{x^{5}} have the same denominator, add them by adding their numerators.

$$\frac{\frac{-1+3x^{3}}{x^{5}}-\frac{1}{x^{6}}}{1+\frac{3}{x^{5}}+\frac{2}{x^{2}}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x^{5} and x^{6} is x^{6}. Multiply \frac{-1+3x^{3}}{x^{5}} times \frac{x}{x}.

$$\frac{\frac{\left(-1+3x^{3}\right)x}{x^{6}}-\frac{1}{x^{6}}}{1+\frac{3}{x^{5}}+\frac{2}{x^{2}}}$$

Since \frac{\left(-1+3x^{3}\right)x}{x^{6}} and \frac{1}{x^{6}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{\left(-1+3x^{3}\right)x-1}{x^{6}}}{1+\frac{3}{x^{5}}+\frac{2}{x^{2}}}$$

Do the multiplications in \left(-1+3x^{3}\right)x-1.

$$\frac{\frac{-x+3x^{4}-1}{x^{6}}}{1+\frac{3}{x^{5}}+\frac{2}{x^{2}}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{x^{5}}{x^{5}}.

$$\frac{\frac{-x+3x^{4}-1}{x^{6}}}{\frac{x^{5}}{x^{5}}+\frac{3}{x^{5}}+\frac{2}{x^{2}}}$$

Since \frac{x^{5}}{x^{5}} and \frac{3}{x^{5}} have the same denominator, add them by adding their numerators.

$$\frac{\frac{-x+3x^{4}-1}{x^{6}}}{\frac{x^{5}+3}{x^{5}}+\frac{2}{x^{2}}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x^{5} and x^{2} is x^{5}. Multiply \frac{2}{x^{2}} times \frac{x^{3}}{x^{3}}.

$$\frac{\frac{-x+3x^{4}-1}{x^{6}}}{\frac{x^{5}+3}{x^{5}}+\frac{2x^{3}}{x^{5}}}$$

Since \frac{x^{5}+3}{x^{5}} and \frac{2x^{3}}{x^{5}} have the same denominator, add them by adding their numerators.

$$\frac{\frac{-x+3x^{4}-1}{x^{6}}}{\frac{x^{5}+3+2x^{3}}{x^{5}}}$$

Divide \frac{-x+3x^{4}-1}{x^{6}} by \frac{x^{5}+3+2x^{3}}{x^{5}} by multiplying \frac{-x+3x^{4}-1}{x^{6}} by the reciprocal of \frac{x^{5}+3+2x^{3}}{x^{5}}.

$$\frac{\left(-x+3x^{4}-1\right)x^{5}}{x^{6}\left(x^{5}+3+2x^{3}\right)}$$

Cancel out x^{5} in both numerator and denominator.

$$\frac{3x^{4}-x-1}{x\left(x^{5}+2x^{3}+3\right)}$$

Use the distributive property to multiply x by x^{5}+2x^{3}+3.

$$\frac{3x^{4}-x-1}{x^{6}+2x^{4}+3x}$$

Expand

$$\frac{3x^{4}-x-1}{x\left(x^{5}+2x^{3}+3\right)}$$

Short Solution Steps

Rewrite x^{6} as xx^{5}. Cancel out x in both numerator and denominator.

$$\frac{-\frac{1}{x^{5}}+\frac{3}{x^{2}}-\frac{1}{x^{6}}}{1+\frac{3}{x^{5}}+\frac{2}{x^{2}}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x^{5} and x^{2} is x^{5}. Multiply \frac{3}{x^{2}} times \frac{x^{3}}{x^{3}}.

$$\frac{-\frac{1}{x^{5}}+\frac{3x^{3}}{x^{5}}-\frac{1}{x^{6}}}{1+\frac{3}{x^{5}}+\frac{2}{x^{2}}}$$

Since -\frac{1}{x^{5}} and \frac{3x^{3}}{x^{5}} have the same denominator, add them by adding their numerators.

$$\frac{\frac{-1+3x^{3}}{x^{5}}-\frac{1}{x^{6}}}{1+\frac{3}{x^{5}}+\frac{2}{x^{2}}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x^{5} and x^{6} is x^{6}. Multiply \frac{-1+3x^{3}}{x^{5}} times \frac{x}{x}.

$$\frac{\frac{\left(-1+3x^{3}\right)x}{x^{6}}-\frac{1}{x^{6}}}{1+\frac{3}{x^{5}}+\frac{2}{x^{2}}}$$

Since \frac{\left(-1+3x^{3}\right)x}{x^{6}} and \frac{1}{x^{6}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{\left(-1+3x^{3}\right)x-1}{x^{6}}}{1+\frac{3}{x^{5}}+\frac{2}{x^{2}}}$$

Do the multiplications in \left(-1+3x^{3}\right)x-1.

$$\frac{\frac{-x+3x^{4}-1}{x^{6}}}{1+\frac{3}{x^{5}}+\frac{2}{x^{2}}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{x^{5}}{x^{5}}.

$$\frac{\frac{-x+3x^{4}-1}{x^{6}}}{\frac{x^{5}}{x^{5}}+\frac{3}{x^{5}}+\frac{2}{x^{2}}}$$

Since \frac{x^{5}}{x^{5}} and \frac{3}{x^{5}} have the same denominator, add them by adding their numerators.

$$\frac{\frac{-x+3x^{4}-1}{x^{6}}}{\frac{x^{5}+3}{x^{5}}+\frac{2}{x^{2}}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x^{5} and x^{2} is x^{5}. Multiply \frac{2}{x^{2}} times \frac{x^{3}}{x^{3}}.

$$\frac{\frac{-x+3x^{4}-1}{x^{6}}}{\frac{x^{5}+3}{x^{5}}+\frac{2x^{3}}{x^{5}}}$$

Since \frac{x^{5}+3}{x^{5}} and \frac{2x^{3}}{x^{5}} have the same denominator, add them by adding their numerators.

$$\frac{\frac{-x+3x^{4}-1}{x^{6}}}{\frac{x^{5}+3+2x^{3}}{x^{5}}}$$

Divide \frac{-x+3x^{4}-1}{x^{6}} by \frac{x^{5}+3+2x^{3}}{x^{5}} by multiplying \frac{-x+3x^{4}-1}{x^{6}} by the reciprocal of \frac{x^{5}+3+2x^{3}}{x^{5}}.

$$\frac{\left(-x+3x^{4}-1\right)x^{5}}{x^{6}\left(x^{5}+3+2x^{3}\right)}$$

Cancel out x^{5} in both numerator and denominator.

$$\frac{3x^{4}-x-1}{x\left(x^{5}+2x^{3}+3\right)}$$

Use the distributive property to multiply x by x^{5}+2x^{3}+3.

$$\frac{3x^{4}-x-1}{x^{6}+2x^{4}+3x}$$