Question

If sum and product of zeroes of a polynomial are ($-3$) and ($-2$) respectively, then a polynomial is

• A. $x^2 - 3x - 2$
• B. $-x^2 - 3x + 2$
• C. $-x^2 + 3x - 2$
• D. $x^2 + 3x + 2$

Hint:

Do not forget that any constant multiple $k(x^2 - Sx + P)$ is a valid polynomial with the same zeroes.
If your initial expression $x^2 + 3x - 2$ is not listed, try multiplying the entire expression by $-1$ to see if its negative counterpart is present!

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
We are given that the sum of the zeroes ($S$) of a quadratic polynomial is $-3$ and the product of the zeroes ($P$) is $-2$.
We need to identify the correct polynomial expression among the options.

Step 2: Key Formula or Approach:
A quadratic polynomial with sum of zeroes $S$ and product of zeroes $P$ can be written as:
\[ p(x) = k(x^2 - Sx + P) \]
where $k$ is any non-zero real constant.
By substituting the given values of $S$ and $P$, we can find the general equation of the polynomial and match it with the options.

Step 3: Detailed Explanation:

• Identify the given parameters:
\[ \text{Sum of zeroes, } S = -3 \]
\[ \text{Product of zeroes, } P = -2 \]

• Substitute these values into the standard quadratic polynomial formula:
\[ p(x) = k(x^2 - (-3)x + (-2)) \]
\[ p(x) = k(x^2 + 3x - 2) \]

• Match with the given options by testing different values of the non-zero real constant $k$:
- If we set $k = 1$:
\[ p(x) = x^2 + 3x - 2 \]
This does not match any of the given options.
- If we set $k = -1$:
\[ p(x) = -(x^2 + 3x - 2) = -x^2 - 3x + 2 \]
This exactly matches option (B).


Step 4: Final Answer:
Therefore, the correct polynomial representation is $-x^2 - 3x + 2$.