Solve b/a^3+1+b/a^3-1

$$\frac { b } { a ^ { 3 } + 1 } + \frac { b } { a ^ { 3 } - 1 }$$

Evaluate

$$\frac{2ba^{3}}{a^{6}-1}$$

$$|a|\neq 1$$

Short Solution Steps

Factor a^{3}+1. Factor a^{3}-1.

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

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

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

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

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

Do the multiplications in b\left(a-1\right)\left(a^{2}+a+1\right)+b\left(a+1\right)\left(a^{2}-a+1\right).

$$\frac{ba^{3}+ba^{2}+ba-ba^{2}-ba-b+ba^{3}-ba^{2}+ba+ba^{2}-ba+b}{\left(a-1\right)\left(a+1\right)\left(a^{2}+a+1\right)\left(a^{2}-a+1\right)}$$

Combine like terms in ba^{3}+ba^{2}+ba-ba^{2}-ba-b+ba^{3}-ba^{2}+ba+ba^{2}-ba+b.

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

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

$$\frac{2ba^{3}}{a^{6}-1}$$

Factor

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