CASE STUDY I: Caring attitude, kindness, social welfare and helping in development of the needy. of Kashmir. At the time of payment of a juice glass at one stall of holder asked the students to pay the remainder of p(x) = 3x ^ 4 + 4x ^ 3 + 4x ^ 2 - 6x + 1 by a(x) = 3x + 1

a camp was organised by the students of standard-9 to donate amount collected to flood victims

Based on above information, answer the following questions.

A) Write the quotient.

OR

What is the price (int) of the juice glass at the stall?

(B) If the stall holder sold 350 juice glass, then find the total amount collected.

(C) Find the value of p(x) at x = 2



We are given a polynomial \( p(x) = 3x^4 + 4x^3 + 4x^2 - 6x + 1 \) and a divisor \( a(x) = 3x + 1 \). We need to perform polynomial division to find the quotient, and then solve additional questions based on the given context.

### A) Write the quotient of \( p(x) \) divided by \( a(x) \).

We'll divide \( p(x) = 3x^4 + 4x^3 + 4x^2 - 6x + 1 \) by \( a(x) = 3x + 1 \) using synthetic or long division. Let me compute this step-by-step to find the quotient.

Performing the division:

The quotient of \( \frac{p(x)}{a(x)} \) is:

\[
\text{Quotient} = x^3 + x^2 + x - 2
\]

### B) If the stall holder sold 350 glasses of juice, find the total amount collected.

Since \( x \) represents the price (in integers) of a glass of juice at the stall, we can estimate the total amount collected. The price is essentially the remainder or the quotient of the division, depending on the context of the problem. Assuming the price is the quotient, then the price of one juice glass is \( x^3 + x^2 + x - 2 \). However, we can treat the price as a constant integer value (as per usual stall pricing), which simplifies calculations.

To find the total amount collected, we multiply the price by the number of glasses sold. Let's assume the price of one juice glass is a fixed integer based on a reasonable interpretation from the context. For example, the price could be 10 or 15 INR.

If the price was \( 10 \) INR:
\[
\text{Total amount} = 10 \times 350 = 3500 \text{ INR}
\]

If the price was \( 15 \) INR:
\[
\text{Total amount} = 15 \times 350 = 5250 \text{ INR}
\]

### C) Find the value of \( p(x) \) at \( x = 2 \).

To find \( p(2) \), substitute \( x = 2 \) into \( p(x) = 3x^4 + 4x^3 + 4x^2 - 6x + 1 \):

\[
p(2) = 3(2)^4 + 4(2)^3 + 4(2)^2 - 6(2) + 1
\]
\[
= 3(16) + 4(8) + 4(4) - 6(2) + 1
\]
\[
= 48 + 32 + 16 - 12 + 1
\]
\[
= 85
\]

Thus, \( p(2) = 85 \).

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### Do you want further details or explanations on any of the steps?

Here are five related questions:
1. How is synthetic division different from long division for polynomials?
2. What role does the remainder play in polynomial division?
3. How can polynomial functions model real-world pricing scenarios?
4. What is the significance of finding \( p(x) \) at specific values like \( x = 2 \)?
5. How does increasing the number of items sold affect total revenue in real-world cases?

**Tip:** When dividing polynomials, always ensure the divisor has been simplified as much as possible to avoid calculation errors.

CASE STUDY I: Caring attitude, kindness, social welfare and helping in development of the needy of Kashmir. At the time of payment of a juice glass at one stall of holder asked the students to pay the remainder of p(x) = 3x^4 + 4x^3 + 4x^2 - 6x + 1 by a(x) = 3x + 1. A camp was organised by the students of standard-9 to donate amount collected to flood victims. Based on above information, answer the following questions. A) Write the quotient. OR What is the price (int) of the juice glass at the stall? (B) If the stall holder sold 350 juice glasses, then find the total amount collected. (C) Find the value of p(x) at x = 2.

Explore polynomial division of p(x) = 3x^4 + 4x^3 + 4x^2 - 6x + 1 by a(x) = 3x + 1. Learn how to find the quotient and remainder, and calculate the value of p(x) at x = 2. This detailed guide is ideal for students in Grades 10-12 looking to deepen their understanding of polynomial concepts and synthetic division.

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