The number of polynomials having zeroes as-2 and 5 is

Options

A. 1
B. 2
C. 3
D. more than 3

Answer

To find the number of polynomials having zeros at -2 and 5, we can follow these steps:

### Step 1: Identify the zeros
The zeros of the polynomial are given as:
- α = -2
- β = 5

### Step 2: Calculate the sum and product of the zeros
The sum of the zeros (α + β) is:
\[ \text{Sum} = -2 + 5 = 3 \]

The product of the zeros (α * β) is:
\[ \text{Product} = -2 \times 5 = -10 \]

### Step 3: Form the polynomial using the sum and product
The general form of a quadratic polynomial with zeros α and β is given by:
\[ P(x) = x^2 - (\text{Sum})x + (\text{Product}) \]
Substituting the values we calculated:
\[ P(x) = x^2 - 3x - 10 \]

### Step 4: Consider the effect of multiplying by a non-zero constant
A polynomial can be multiplied by any non-zero constant, and it will still have the same zeros. For example, if we multiply the polynomial by a constant \( k \) (where \( k \neq 0 \)):
\[ P(x) = k(x^2 - 3x - 10) \]
This will still have the zeros at -2 and 5.

### Step 5: Conclusion on the number of polynomials
Since we can choose any non-zero constant \( k \), there are infinitely many polynomials that can be formed with the zeros -2 and 5.

### Final Answer
Thus, the number of polynomials having zeros as -2 and 5 is infinite.

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