Number of Polynomials Having Given Zeroes

Video Explanation

Question

The number of polynomials having zeroes \( -2 \) and \( 5 \) is:

Solution

Step 1: Form a Polynomial with the Given Zeroes

If a polynomial has zeroes \( -2 \) and \( 5 \), then

\[ (x + 2)(x – 5) \]

is a factor of the polynomial.

Step 2: Introduce a Non-zero Constant

Any non-zero constant multiple of this factor will also have the same zeroes.

So, the general polynomial is:

\[ k(x + 2)(x – 5), \quad \text{where } k \neq 0 \]

Step 3: Count the Number of Such Polynomials

Since \(k\) can take infinitely many non-zero real values,

there are infinitely many such polynomials.

Conclusion

The number of polynomials having zeroes \( -2 \) and \( 5 \) is:

\[ \boxed{\text{Infinitely many}} \]