Question

A polynomial \(p(x)\), which has sum of its zeroes equal to their product, is :

• A. \(3x^2 + 2x + 2\)
• B. \(3x^2 - 2x - 3\)
• C. \(3x^2 - 2x + 2\)
• D. \(x^2 - 3x + 2\)

Hint:

For any quadratic polynomial \( ax^2 + bx + c \), the condition "sum of zeroes equals product of zeroes" simplifies to \( b + c = 0 \).
You can quickly identify the correct polynomial by simply adding the coefficient of \( x \) and the constant term to see if they sum up to zero.
For Option (C), \( -2 + 2 = 0 \), giving you the answer instantly.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The topic of this question is Quadratic Polynomials.
We are looking for a quadratic polynomial \( p(x) = ax^2 + bx + c \) such that the sum of its zeroes is equal to the product of its zeroes.

Step 2: Key Formula or Approach:
For a general quadratic polynomial \( ax^2 + bx + c \) with zeroes \( \alpha \) and \( \beta \):
- Sum of zeroes is given by: \( \alpha + \beta = -\frac{b}{a} \)
- Product of zeroes is given by: \( \alpha\beta = \frac{c}{a} \)
According to the given condition, the sum of the zeroes is equal to the product of the zeroes:
\[ -\frac{b}{a} = \frac{c}{a} \]
Since \( a \neq 0 \) for a quadratic polynomial, we can multiply both sides by \( a \) to get:
\[ -b = c \quad \text{or} \quad b + c = 0 \]

Step 3: Detailed Explanation:
Let us test the condition \( -b = c \) (or \( b + c = 0 \)) for each of the given options:
1. For Option (A) \( 3x^2 + 2x + 2 \):
Here, \( a = 3, b = 2, c = 2 \).
Checking the sum: \( b + c = 2 + 2 = 4 \neq 0 \).
Sum of zeroes \( = -\frac{2}{3} \), and Product of zeroes \( = \frac{2}{3} \). They are not equal.
2. For Option (B) \( 3x^2 - 2x - 3 \):
Here, \( a = 3, b = -2, c = -3 \).
Checking the sum: \( b + c = -2 + (-3) = -5 \neq 0 \).
Sum of zeroes \( = -\frac{-2}{3} = \frac{2}{3} \), and Product of zeroes \( = \frac{-3}{3} = -1 \). They are not equal.
3. For Option (C) \( 3x^2 - 2x + 2 \):
Here, \( a = 3, b = -2, c = 2 \).
Checking the sum: \( b + c = -2 + 2 = 0 \).
This satisfies our condition.
Let's verify: Sum of zeroes \( = -\frac{-2}{3} = \frac{2}{3} \), and Product of zeroes \( = \frac{2}{3} \).
Both values are equal, so this option is correct.
4. For Option (D) \( x^2 - 3x + 2 \):
Here, \( a = 1, b = -3, c = 2 \).
Checking the sum: \( b + c = -3 + 2 = -1 \neq 0 \).
Sum of zeroes \( = -\frac{-3}{1} = 3 \), and Product of zeroes \( = \frac{2}{1} = 2 \). They are not equal.

Step 4: Final Answer:
The polynomial that has the sum of its zeroes equal to their product is \(3x^2 - 2x + 2\), which is option (C).