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

Options

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

Answer

To solve the problem where \( R = \frac{a^3 + 1}{a - 1} \) and \( S = \frac{a^2 - a + 1}{a - 1} \), we need to find \( R \div S \). Here’s the step-by-step solution:

### Step 1: Write the expression for \( R \div S \)
We start with the expression:
\[ R \div S = \frac{R}{S} = \frac{\frac{a^3 + 1}{a - 1}}{\frac{a^2 - a + 1}{a - 1}} \]

### Step 2: Change the division to multiplication
To simplify the division of fractions, we multiply by the reciprocal:
\[ R \div S = \frac{a^3 + 1}{a - 1} \times \frac{a - 1}{a^2 - a + 1} \]

### Step 3: Cancel common terms
The \( a - 1 \) in the numerator and denominator cancels out:
\[ R \div S = \frac{a^3 + 1}{a^2 - a + 1} \]

### Step 4: Factor \( a^3 + 1 \)
We can factor \( a^3 + 1 \) using the sum of cubes formula:
\[ a^3 + 1 = (a + 1)(a^2 - a + 1) \]

### Step 5: Substitute the factored form
Now we substitute this back into our expression:
\[ R \div S = \frac{(a + 1)(a^2 - a + 1)}{a^2 - a + 1} \]

### Step 6: Cancel the common factor
The \( a^2 - a + 1 \) in the numerator and denominator cancels out:
\[ R \div S = a + 1 \]

### Final Answer
Thus, the value of \( R \div S \) is:
\[ \boxed{a + 1} \]