NCERT Exemplar Solutions
Class 10 - Mathematics - CHAPTER 2: Polynomials - Exercise 2.1

Question 2

Step 1: Write the factor form of the polynomial.

If the zeroes are \(-3\) and \(4\), then the polynomial must be a multiple of

\[(x - (-3))(x - 4) = (x + 3)(x - 4)\]

Step 2: Expand the product.

\[(x + 3)(x - 4) = x^2 - 4x + 3x - 12\]

\[= x^2 - x - 12\]

Step 3: Compare with the given options.

The basic polynomial is \(x^2 - x - 12\).

Any non-zero constant multiple of this is also valid.

Step 4: Check the options.

• Option A: \(x^2 - x + 12\) → wrong (constant term differs).
• Option B: \(x^2 + x + 12\) → wrong (signs do not match).
• Option C: \(\dfrac{1}{2}(x^2 - x - 12) = \dfrac{x^2}{2} - \dfrac{x}{2} - 6\) → correct.
• Option D: \(2x^2 + 2x - 24\) → wrong, because it equals \(2(x^2 + x - 12)\), not a multiple of \(x^2 - x - 12\).

Conclusion: The correct polynomial is

\(\dfrac{x^2}{2} - \dfrac{x}{2} - 6\).