Question

The sum and product of zeroes of a quadratic polynomial $p(x)$ are $-\frac{1}{3}$ and $2$ respectively. The polynomial $p(x)$ is :

• A. $3x^2 - x + 6$
• B. $x^2 + \frac{1}{3}x - 2$
• C. $3x^2 - x + 2$
• D. $-3x^2 - x - 6$

Hint:

For any quadratic polynomial $ax^2 + bx + c$, the sum of zeroes is $-\frac{b}{a}$ and the product is $\frac{c}{a}$.
In a multiple-choice question, you can quickly test this on the options:
For Option (A): Sum $= -\frac{-1}{3} = \frac{1}{3} \neq -\frac{1}{3}$.
For Option (B): Product $= \frac{-2}{1} = -2 \neq 2$.
For Option (C): Sum $= -\frac{-1}{3} = \frac{1}{3} \neq -\frac{1}{3}$.
For Option (D): Sum $= -\frac{-1}{-3} = -\frac{1}{3}$ and Product $= \frac{-6}{-3} = 2$.
This verification process takes only a few seconds and prevents mistakes!

Answer 1

The Correct Option is D

Step 1: Instead of building a general formula, test each option directly using sum = -b/a and product = c/a.
For any quadratic $ax^2+bx+c$, the sum of zeroes is $-\frac{b}{a}$ and the product of zeroes is $\frac{c}{a}$. We need a polynomial where the sum is $-\frac{1}{3}$ and the product is $2$. Let's check each option in turn.

Step 2: Test Option (A), $3x^2 - x + 6$.
Here $a=3, b=-1, c=6$. Sum $= -\frac{-1}{3} = \frac{1}{3}$. This is $+\frac{1}{3}$, not $-\frac{1}{3}$, so this option fails immediately.

Step 3: Test Option (B), $x^2 + \frac{1}{3}x - 2$.
Here $a=1, b=\frac{1}{3}, c=-2$. Sum $= -\frac{1/3}{1} = -\frac{1}{3}$ (matches). Product $= \frac{-2}{1} = -2$. This should be $2$, not $-2$, so this option fails on the product.

Step 4: Test Option (C), $3x^2 - x + 2$.
Here $a=3, b=-1, c=2$. Sum $= -\frac{-1}{3} = \frac{1}{3}$, which is $+\frac{1}{3}$, not $-\frac{1}{3}$. This option fails on the sum.

Step 5: Test Option (D), $-3x^2 - x - 6$.
Here $a=-3, b=-1, c=-6$. Sum $= -\frac{-1}{-3} = -\frac{1}{3}$ (matches). Product $= \frac{-6}{-3} = 2$ (matches). Both conditions hold exactly.

Final Answer:
Only Option (D), $-3x^2 - x - 6$, has sum of zeroes $-\frac{1}{3}$ and product of zeroes $2$, which corresponds to Option (D). \[ \boxed{p(x) = -3x^2 - x - 6} \]