Akshay's father has given him money to buy some apples from the market at the rate of `x^(2) - 5x + 6` per apple. The total amount of money given by his father is represented by `4x^(4) + 2x^(3) - 2x^(2) + x - 1`. Find the maximum amount of money he should have so that he is able to buy exact number of apples from the market.

Options

A. `4x^(4)+2x^(3)-2x^(2) + 288 x + 504 `
B. `4x^(4) + 2x^(3) - 2x^(2) - 288x - 504`
C. `4x^(4) +2x^(3) - 2x^(2) - 288x + 504 `
D. None of these

Answer

To solve the problem, we need to determine the maximum amount of money Akshay should have so that he can buy an exact number of apples. This involves dividing the total amount of money given by his father by the cost per apple and ensuring that there is no remainder.

### Step-by-Step Solution:

1. **Identify the Cost of One Apple**:
The cost of one apple is given by the polynomial:
\[ C(x) = x^2 - 5x + 6 \]

2. **Identify the Total Amount of Money**:
The total amount of money given by Akshay's father is represented by the polynomial:
\[ M(x) = 4x^4 + 2x^3 - 2x^2 + x - 1 \]

3. **Perform Polynomial Division**:
We need to divide \( M(x) \) by \( C(x) \) to find the quotient and the remainder. The goal is to ensure that the remainder is zero for Akshay to buy an exact number of apples.

- **Divide \( M(x) \) by \( C(x) \)**:
Using polynomial long division, we divide \( M(x) \) by \( C(x) \).

- **Step 1**: Divide the leading term of \( M(x) \) by the leading term of \( C(x) \):
\[ \frac{4x^4}{x^2} = 4x^2 \]

- **Step 2**: Multiply \( C(x) \) by \( 4x^2 \):
\[ 4x^2 \cdot (x^2 - 5x + 6) = 4x^4 - 20x^3 + 24x^2 \]

- **Step 3**: Subtract this from \( M(x) \):
\[ (4x^4 + 2x^3 - 2x^2 + x - 1) - (4x^4 - 20x^3 + 24x^2) = 22x^3 - 26x^2 + x - 1 \]

- **Step 4**: Repeat the process with the new polynomial \( 22x^3 - 26x^2 + x - 1 \):
\[ \frac{22x^3}{x^2} = 22x \]
Multiply and subtract:
\[ 22x \cdot (x^2 - 5x + 6) = 22x^3 - 110x^2 + 132x \]
\[ (22x^3 - 26x^2 + x - 1) - (22x^3 - 110x^2 + 132x) = 84x^2 - 131x - 1 \]

- **Step 5**: Repeat again:
\[ \frac{84x^2}{x^2} = 84 \]
Multiply and subtract:
\[ 84 \cdot (x^2 - 5x + 6) = 84x^2 - 420x + 504 \]
\[ (84x^2 - 131x - 1) - (84x^2 - 420x + 504) = 289x - 505 \]

- **Step 6**: The remainder is \( 289x - 505 \).

4. **Set the Remainder to Zero**:
For Akshay to buy an exact number of apples, the remainder must be zero:
\[ 289x - 505 = 0 \]
Solving for \( x \):
\[ 289x = 505 \implies x = \frac{505}{289} \]

5. **Calculate the Maximum Amount of Money**:
Substitute \( x \) back into \( M(x) \) to find the maximum amount of money:
\[ M\left(\frac{505}{289}\right) = 4\left(\frac{505}{289}\right)^4 + 2\left(\frac{505}{289}\right)^3 - 2\left(\frac{505}{289}\right)^2 + \left(\frac{505}{289}\right) - 1 \]
(This calculation will yield the maximum amount of money.)