Solve 3x^2+5x+1/x^3-1-1-x/x^2+x+1-3/x-1

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

Evaluate

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

Short Solution Steps

Factor x^{3}-1.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-1\right)\left(x^{2}+x+1\right) and x^{2}+x+1 is \left(x-1\right)\left(x^{2}+x+1\right). Multiply \frac{1-x}{x^{2}+x+1} times \frac{x-1}{x-1}.

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

Since \frac{3x^{2}+5x+1}{\left(x-1\right)\left(x^{2}+x+1\right)} and \frac{\left(1-x\right)\left(x-1\right)}{\left(x-1\right)\left(x^{2}+x+1\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in 3x^{2}+5x+1-\left(1-x\right)\left(x-1\right).

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

Combine like terms in 3x^{2}+5x+1-x+1+x^{2}-x.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-1\right)\left(x^{2}+x+1\right) and x-1 is \left(x-1\right)\left(x^{2}+x+1\right). Multiply \frac{3}{x-1} times \frac{x^{2}+x+1}{x^{2}+x+1}.

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

Since \frac{4x^{2}+3x+2}{\left(x-1\right)\left(x^{2}+x+1\right)} and \frac{3\left(x^{2}+x+1\right)}{\left(x-1\right)\left(x^{2}+x+1\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in 4x^{2}+3x+2-3\left(x^{2}+x+1\right).

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

Combine like terms in 4x^{2}+3x+2-3x^{2}-3x-3.

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

Factor the expressions that are not already factored in \frac{x^{2}-1}{\left(x-1\right)\left(x^{2}+x+1\right)}.

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

Cancel out x-1 in both numerator and denominator.

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

Expand

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

Short Solution Steps

Factor x^{3}-1.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-1\right)\left(x^{2}+x+1\right) and x^{2}+x+1 is \left(x-1\right)\left(x^{2}+x+1\right). Multiply \frac{1-x}{x^{2}+x+1} times \frac{x-1}{x-1}.

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

Since \frac{3x^{2}+5x+1}{\left(x-1\right)\left(x^{2}+x+1\right)} and \frac{\left(1-x\right)\left(x-1\right)}{\left(x-1\right)\left(x^{2}+x+1\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in 3x^{2}+5x+1-\left(1-x\right)\left(x-1\right).

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

Combine like terms in 3x^{2}+5x+1-x+1+x^{2}-x.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-1\right)\left(x^{2}+x+1\right) and x-1 is \left(x-1\right)\left(x^{2}+x+1\right). Multiply \frac{3}{x-1} times \frac{x^{2}+x+1}{x^{2}+x+1}.

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

Since \frac{4x^{2}+3x+2}{\left(x-1\right)\left(x^{2}+x+1\right)} and \frac{3\left(x^{2}+x+1\right)}{\left(x-1\right)\left(x^{2}+x+1\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in 4x^{2}+3x+2-3\left(x^{2}+x+1\right).

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

Combine like terms in 4x^{2}+3x+2-3x^{2}-3x-3.

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

Factor the expressions that are not already factored in \frac{x^{2}-1}{\left(x-1\right)\left(x^{2}+x+1\right)}.

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

Cancel out x-1 in both numerator and denominator.

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