Vikas has Rs. `(x^(3)+2ax+b)`, with this money he can buy exactly (x-1) jeans or (x+1) shirts with no money left. How much money Vikas has if x=4?

Options

A. Rs. 80
B. Rs. 120
C. Rs. 30
D. Rs. 60

Answer

To solve the problem, we need to find the values of \( a \) and \( b \) in the polynomial \( x^3 + 2ax + b \) given that Vikas can buy \( (x - 1) \) jeans or \( (x + 1) \) shirts with no money left.

### Step-by-Step Solution:

1. **Understanding the Problem**:
Vikas has an amount represented by the polynomial \( x^3 + 2ax + b \). He can buy \( (x - 1) \) jeans or \( (x + 1) \) shirts. This means that both \( (x - 1) \) and \( (x + 1) \) are factors of the polynomial.

2. **Using the Factor Theorem**:
According to the Factor Theorem, if \( (x - c) \) is a factor of a polynomial, then substituting \( x = c \) into the polynomial will yield zero.

3. **Finding the First Equation**:
- For jeans: Set \( x = 1 \) (since \( x - 1 = 0 \)):
\[ 1^3 + 2a(1) + b = 0 \]
Simplifying gives:
\[ 1 + 2a + b = 0 \quad \text{(Equation 1)} \]

4. **Finding the Second Equation**:
- For shirts: Set \( x = -1 \) (since \( x + 1 = 0 \)):
\[ (-1)^3 + 2a(-1) + b = 0 \]
Simplifying gives:
\[ -1 - 2a + b = 0 \quad \text{(Equation 2)} \]

5. **Solving the System of Equations**:
- From Equation 1:
\[ b = -1 - 2a \]
- Substitute \( b \) in Equation 2:
\[ -1 - 2a + (-1 - 2a) = 0 \]
Simplifying gives:
\[ -2 - 4a = 0 \implies 4a = -2 \implies a = -\frac{1}{2} \]
- Substitute \( a \) back into Equation 1 to find \( b \):
\[ b = -1 - 2(-\frac{1}{2}) = -1 + 1 = 0 \]

6. **Final Polynomial**:
Now we have \( a = -\frac{1}{2} \) and \( b = 0 \). Thus, the polynomial becomes:
\[ x^3 - x = x(x^2 - 1) = x(x - 1)(x + 1) \]

7. **Calculating the Amount for \( x = 4 \)**:
Substitute \( x = 4 \) into the polynomial:
\[ 4^3 + 2(-\frac{1}{2})(4) + 0 = 64 - 4 + 0 = 60 \]

### Conclusion:
Vikas has Rs. **60**.