Solve (a + 2)/(a ^ 3 + 2a + 4) - 1/(a - 2) + (a + 2a)/(a ^ 3 - 8) = | 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

$$\frac{a+2}{a^{3}+2a+4}-\frac{1}{a-2}+\frac{a+2a}{a^{3}-8}=$$

Evaluate

$\frac{-a^{5}+2a^{4}-4a^{3}-2a^{2}-12a-32}{\left(a^{3}-8\right)\left(a^{3}+2a+4\right)}$

Short Solution Steps

Combine $a$ and $2a$ to get $3a$.

$$\frac{a+2}{a^{3}+2a+4}-\frac{1}{a-2}+\frac{3a}{a^{3}-8}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $a^{3}+2a+4$ and $a-2$ is $\left(a-2\right)\left(a^{3}+2a+4\right)$. Multiply $\frac{a+2}{a^{3}+2a+4}$ times $\frac{a-2}{a-2}$. Multiply $\frac{1}{a-2}$ times $\frac{a^{3}+2a+4}{a^{3}+2a+4}$.

$$\frac{\left(a+2\right)\left(a-2\right)}{\left(a-2\right)\left(a^{3}+2a+4\right)}-\frac{a^{3}+2a+4}{\left(a-2\right)\left(a^{3}+2a+4\right)}+\frac{3a}{a^{3}-8}$$

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

$$\frac{\left(a+2\right)\left(a-2\right)-\left(a^{3}+2a+4\right)}{\left(a-2\right)\left(a^{3}+2a+4\right)}+\frac{3a}{a^{3}-8}$$

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

$$\frac{a^{2}-2a+2a-4-a^{3}-2a-4}{\left(a-2\right)\left(a^{3}+2a+4\right)}+\frac{3a}{a^{3}-8}$$

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

$$\frac{a^{2}-2a-8-a^{3}}{\left(a-2\right)\left(a^{3}+2a+4\right)}+\frac{3a}{a^{3}-8}$$

Factor $a^{3}-8$.

$$\frac{a^{2}-2a-8-a^{3}}{\left(a-2\right)\left(a^{3}+2a+4\right)}+\frac{3a}{\left(a-2\right)\left(a^{2}+2a+4\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $\left(a-2\right)\left(a^{3}+2a+4\right)$ and $\left(a-2\right)\left(a^{2}+2a+4\right)$ is $\left(a-2\right)\left(a^{2}+2a+4\right)\left(a^{3}+2a+4\right)$. Multiply $\frac{a^{2}-2a-8-a^{3}}{\left(a-2\right)\left(a^{3}+2a+4\right)}$ times $\frac{a^{2}+2a+4}{a^{2}+2a+4}$. Multiply $\frac{3a}{\left(a-2\right)\left(a^{2}+2a+4\right)}$ times $\frac{a^{3}+2a+4}{a^{3}+2a+4}$.

$$\frac{\left(a^{2}-2a-8-a^{3}\right)\left(a^{2}+2a+4\right)}{\left(a-2\right)\left(a^{2}+2a+4\right)\left(a^{3}+2a+4\right)}+\frac{3a\left(a^{3}+2a+4\right)}{\left(a-2\right)\left(a^{2}+2a+4\right)\left(a^{3}+2a+4\right)}$$

Since $\frac{\left(a^{2}-2a-8-a^{3}\right)\left(a^{2}+2a+4\right)}{\left(a-2\right)\left(a^{2}+2a+4\right)\left(a^{3}+2a+4\right)}$ and $\frac{3a\left(a^{3}+2a+4\right)}{\left(a-2\right)\left(a^{2}+2a+4\right)\left(a^{3}+2a+4\right)}$ have the same denominator, add them by adding their numerators.

$$\frac{\left(a^{2}-2a-8-a^{3}\right)\left(a^{2}+2a+4\right)+3a\left(a^{3}+2a+4\right)}{\left(a-2\right)\left(a^{2}+2a+4\right)\left(a^{3}+2a+4\right)}$$

Do the multiplications in $\left(a^{2}-2a-8-a^{3}\right)\left(a^{2}+2a+4\right)+3a\left(a^{3}+2a+4\right)$.

$$\frac{a^{4}+2a^{3}+4a^{2}-2a^{3}-4a^{2}-8a-8a^{2}-16a-32-a^{5}-2a^{4}-4a^{3}+3a^{4}+6a^{2}+12a}{\left(a-2\right)\left(a^{2}+2a+4\right)\left(a^{3}+2a+4\right)}$$

Combine like terms in $a^{4}+2a^{3}+4a^{2}-2a^{3}-4a^{2}-8a-8a^{2}-16a-32-a^{5}-2a^{4}-4a^{3}+3a^{4}+6a^{2}+12a$.

$$\frac{2a^{4}-4a^{3}-2a^{2}-12a-32-a^{5}}{\left(a-2\right)\left(a^{2}+2a+4\right)\left(a^{3}+2a+4\right)}$$

Expand $\left(a-2\right)\left(a^{2}+2a+4\right)\left(a^{3}+2a+4\right)$.

$$\frac{2a^{4}-4a^{3}-2a^{2}-12a-32-a^{5}}{a^{6}+2a^{4}-4a^{3}-16a-32}$$

Expand

$\frac{-a^{5}+2a^{4}-4a^{3}-2a^{2}-12a-32}{\left(a^{3}-8\right)\left(a^{3}+2a+4\right)}$

Short Solution Steps

Combine $a$ and $2a$ to get $3a$.

$$\frac{a+2}{a^{3}+2a+4}-\frac{1}{a-2}+\frac{3a}{a^{3}-8}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $a^{3}+2a+4$ and $a-2$ is $\left(a-2\right)\left(a^{3}+2a+4\right)$. Multiply $\frac{a+2}{a^{3}+2a+4}$ times $\frac{a-2}{a-2}$. Multiply $\frac{1}{a-2}$ times $\frac{a^{3}+2a+4}{a^{3}+2a+4}$.

$$\frac{\left(a+2\right)\left(a-2\right)}{\left(a-2\right)\left(a^{3}+2a+4\right)}-\frac{a^{3}+2a+4}{\left(a-2\right)\left(a^{3}+2a+4\right)}+\frac{3a}{a^{3}-8}$$

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

$$\frac{\left(a+2\right)\left(a-2\right)-\left(a^{3}+2a+4\right)}{\left(a-2\right)\left(a^{3}+2a+4\right)}+\frac{3a}{a^{3}-8}$$

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

$$\frac{a^{2}-2a+2a-4-a^{3}-2a-4}{\left(a-2\right)\left(a^{3}+2a+4\right)}+\frac{3a}{a^{3}-8}$$

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

$$\frac{a^{2}-2a-8-a^{3}}{\left(a-2\right)\left(a^{3}+2a+4\right)}+\frac{3a}{a^{3}-8}$$

Factor $a^{3}-8$.

$$\frac{a^{2}-2a-8-a^{3}}{\left(a-2\right)\left(a^{3}+2a+4\right)}+\frac{3a}{\left(a-2\right)\left(a^{2}+2a+4\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $\left(a-2\right)\left(a^{3}+2a+4\right)$ and $\left(a-2\right)\left(a^{2}+2a+4\right)$ is $\left(a-2\right)\left(a^{2}+2a+4\right)\left(a^{3}+2a+4\right)$. Multiply $\frac{a^{2}-2a-8-a^{3}}{\left(a-2\right)\left(a^{3}+2a+4\right)}$ times $\frac{a^{2}+2a+4}{a^{2}+2a+4}$. Multiply $\frac{3a}{\left(a-2\right)\left(a^{2}+2a+4\right)}$ times $\frac{a^{3}+2a+4}{a^{3}+2a+4}$.

$$\frac{\left(a^{2}-2a-8-a^{3}\right)\left(a^{2}+2a+4\right)}{\left(a-2\right)\left(a^{2}+2a+4\right)\left(a^{3}+2a+4\right)}+\frac{3a\left(a^{3}+2a+4\right)}{\left(a-2\right)\left(a^{2}+2a+4\right)\left(a^{3}+2a+4\right)}$$

Since $\frac{\left(a^{2}-2a-8-a^{3}\right)\left(a^{2}+2a+4\right)}{\left(a-2\right)\left(a^{2}+2a+4\right)\left(a^{3}+2a+4\right)}$ and $\frac{3a\left(a^{3}+2a+4\right)}{\left(a-2\right)\left(a^{2}+2a+4\right)\left(a^{3}+2a+4\right)}$ have the same denominator, add them by adding their numerators.

$$\frac{\left(a^{2}-2a-8-a^{3}\right)\left(a^{2}+2a+4\right)+3a\left(a^{3}+2a+4\right)}{\left(a-2\right)\left(a^{2}+2a+4\right)\left(a^{3}+2a+4\right)}$$

Do the multiplications in $\left(a^{2}-2a-8-a^{3}\right)\left(a^{2}+2a+4\right)+3a\left(a^{3}+2a+4\right)$.

$$\frac{a^{4}+2a^{3}+4a^{2}-2a^{3}-4a^{2}-8a-8a^{2}-16a-32-a^{5}-2a^{4}-4a^{3}+3a^{4}+6a^{2}+12a}{\left(a-2\right)\left(a^{2}+2a+4\right)\left(a^{3}+2a+4\right)}$$

Combine like terms in $a^{4}+2a^{3}+4a^{2}-2a^{3}-4a^{2}-8a-8a^{2}-16a-32-a^{5}-2a^{4}-4a^{3}+3a^{4}+6a^{2}+12a$.

$$\frac{2a^{4}-4a^{3}-2a^{2}-12a-32-a^{5}}{\left(a-2\right)\left(a^{2}+2a+4\right)\left(a^{3}+2a+4\right)}$$

Expand $\left(a-2\right)\left(a^{2}+2a+4\right)\left(a^{3}+2a+4\right)$.

$$\frac{2a^{4}-4a^{3}-2a^{2}-12a-32-a^{5}}{a^{6}+2a^{4}-4a^{3}-16a-32}$$