Priya has Rs. `(x^(3) + x^(2) - 17x + 20)`.She wants to buy ice-cream cones each of cost Rs. `(x-3)`. After buying maximum number of ice-cream cones with her money, how much money is left with her?

Options

A. Rs. 10
B. Rs. 50
C. Rs. 15
D. Rs. 5

Answer

To solve the problem step by step, we need to find out how much money Priya has left after buying the maximum number of ice-cream cones.

### Step 1: Understand the problem
Priya has Rs. \( x^3 + x^2 - 17x + 20 \) and each ice-cream cone costs Rs. \( x - 3 \). We need to determine how many cones she can buy and how much money will be left after her purchases.

### Step 2: Set up the equation
Let \( y \) be the number of ice-cream cones Priya buys. The total cost for \( y \) cones is given by:
\[ \text{Total Cost} = y \cdot (x - 3) \]
Since Priya cannot spend more than she has, we can set up the inequality:
\[ y \cdot (x - 3) \leq x^3 + x^2 - 17x + 20 \]

### Step 3: Solve for \( y \)
To find the maximum number of cones she can buy, we can express \( y \) as:
\[ y \leq \frac{x^3 + x^2 - 17x + 20}{x - 3} \]
Next, we need to perform polynomial long division to simplify this expression.

### Step 4: Polynomial long division
Divide \( x^3 + x^2 - 17x + 20 \) by \( x - 3 \):

1. Divide the leading term: \( x^3 \div x = x^2 \).
2. Multiply \( x^2 \) by \( x - 3 \): \( x^3 - 3x^2 \).
3. Subtract:
\[ (x^3 + x^2) - (x^3 - 3x^2) = 4x^2 \]
4. Bring down the next term: \( 4x^2 - 17x \).
5. Divide the leading term: \( 4x^2 \div x = 4x \).
6. Multiply \( 4x \) by \( x - 3 \): \( 4x^2 - 12x \).
7. Subtract:
\[ (4x^2 - 17x) - (4x^2 - 12x) = -5x \]
8. Bring down the next term: \( -5x + 20 \).
9. Divide the leading term: \( -5x \div x = -5 \).
10. Multiply \( -5 \) by \( x - 3 \): \( -5x + 15 \).
11. Subtract:
\[ (-5x + 20) - (-5x + 15) = 5 \]

Thus, the result of the division is:
\[ x^2 + 4x - 5 + \frac{5}{x - 3} \]

### Step 5: Determine the maximum number of cones
The maximum number of cones \( y \) is given by:
\[ y = x^2 + 4x - 5 \]
since \( \frac{5}{x - 3} \) is a fraction and cannot contribute to the number of cones.

### Step 6: Calculate the total cost of the cones
The total cost for \( y \) cones is:
\[ \text{Total Cost} = (x^2 + 4x - 5)(x - 3) \]

### Step 7: Calculate the remaining money
The remaining money after buying the cones is:
\[ \text{Remaining Money} = \text{Total Money} - \text{Total Cost} \]
Substituting the values:
\[ \text{Remaining Money} = (x^3 + x^2 - 17x + 20) - [(x^2 + 4x - 5)(x - 3)] \]

### Step 8: Simplify the remaining money
Calculating the total cost:
\[ (x^2 + 4x - 5)(x - 3) = x^3 + 4x^2 - 5x - 3x^2 - 12x + 15 = x^3 + (4x^2 - 3x^2) + (-5x - 12x) + 15 = x^3 + x^2 - 17x + 15 \]
Now substituting back:
\[ \text{Remaining Money} = (x^3 + x^2 - 17x + 20) - (x^3 + x^2 - 17x + 15) = 20 - 15 = 5 \]

### Final Answer
The amount of money left with Priya after buying the maximum number of ice-cream cones is Rs. 5.