If `R = (a^(3)+1)/(a-1)` and `S = (a^(2) - a+1)/(a+1)`, then R + S is:

Options

A. `(a-1)/(a+1)`
B. `1/(a^(2) + a+1)`
C. 1
D. `(a(a^3+2a^2-2a+3))/(a^2-1)`

Answer

To solve the problem, we need to find \( R + S \) where:

\[ R = \frac{a^3 + 1}{a - 1} \]
\[ S = \frac{a^2 - a + 1}{a + 1} \]

### Step 1: Simplifying \( R \)

First, we can factor \( a^3 + 1 \) using the sum of cubes formula:

\[ a^3 + 1 = (a + 1)(a^2 - a + 1) \]

Thus, we can rewrite \( R \):

\[ R = \frac{(a + 1)(a^2 - a + 1)}{a - 1} \]

### Step 2: Finding a common denominator for \( R + S \)

Next, we need to add \( R \) and \( S \). The common denominator will be \( (a - 1)(a + 1) \):

\[ R + S = \frac{(a + 1)(a^2 - a + 1)}{a - 1} + \frac{a^2 - a + 1}{a + 1} \]

### Step 3: Rewrite \( S \) with the common denominator

We rewrite \( S \) to have the common denominator:

\[ S = \frac{(a^2 - a + 1)(a - 1)}{(a + 1)(a - 1)} \]

### Step 4: Combine the two fractions

Now we can combine \( R \) and \( S \):

\[ R + S = \frac{(a + 1)(a^2 - a + 1) + (a^2 - a + 1)(a - 1)}{(a - 1)(a + 1)} \]

### Step 5: Simplifying the numerator

Now we simplify the numerator:

1. Expand \( (a + 1)(a^2 - a + 1) \):
\[ = a^3 - a^2 + a + a^2 - a + 1 = a^3 + 1 \]

2. Expand \( (a^2 - a + 1)(a - 1) \):
\[ = a^3 - a^2 + a - a^2 + a - 1 = a^3 - 2a^2 + 2a - 1 \]

Now, combine both expansions:

\[ a^3 + 1 + a^3 - 2a^2 + 2a - 1 = 2a^3 - 2a^2 + 2a \]

### Step 6: Final expression for \( R + S \)

Thus, we have:

\[ R + S = \frac{2a^3 - 2a^2 + 2a}{(a - 1)(a + 1)} \]

### Step 7: Factor out the numerator

We can factor out \( 2 \):

\[ R + S = \frac{2(a^3 - a^2 + a)}{(a - 1)(a + 1)} \]

### Step 8: Final Result

The final expression for \( R + S \) is:

\[ R + S = \frac{2a(a^2 - a + 1)}{(a - 1)(a + 1)} \]