Solve (2x)/(x-1)+(2(x)^(3)+2(x)^(2))/(1-(x)^(3))+(1)/((x)^(2)+x+1) | Equatix - Math Solver

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Example

3(r+2s)=2t-4

a⋅(b-2)=3b

3(r+2s)=2t-4

a⋅(b-2)=3b

Question

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

Evaluate

$\frac{3x-1}{x^{3}-1}$

Short Solution Steps

Factor $1-x^{3}$.

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

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

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

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

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

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

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

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

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

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

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

Since $\frac{-\left(-2\right)x}{\left(x-1\right)\left(x^{2}+x+1\right)}$ and $\frac{x-1}{\left(x-1\right)\left(x^{2}+x+1\right)}$ have the same denominator, add them by adding their numerators.

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

Do the multiplications in $-\left(-2\right)x+x-1$.

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

Combine like terms in $2x+x-1$.

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

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

$$\frac{3x-1}{x^{3}-1}$$

Expand

$-\frac{1-3x}{x^{3}-1}$

Short Solution Steps

Factor $1-x^{3}$.

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

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

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

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

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

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

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

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

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

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

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

Since $\frac{-\left(-2\right)x}{\left(x-1\right)\left(x^{2}+x+1\right)}$ and $\frac{x-1}{\left(x-1\right)\left(x^{2}+x+1\right)}$ have the same denominator, add them by adding their numerators.

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

Do the multiplications in $-\left(-2\right)x+x-1$.

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

Combine like terms in $2x+x-1$.

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

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

$$\frac{3x-1}{x^{3}-1}$$