The number of polynomials having zeroes −2 and 5

Video Explanation

Watch the video explanation below:

Given

The zeroes of the polynomial are:

−2 and 5

To Find

The number of polynomials having these zeroes.

Solution

If −2 and 5 are the zeroes of a polynomial, then the simplest polynomial having these zeroes is:

(x + 2)(x − 5)

= x² − 3x − 10

Now, any non-zero constant multiplied with this polynomial will also have the same zeroes.

So, the general polynomial having zeroes −2 and 5 is:

k(x + 2)(x − 5), where k ≠ 0

Since k can take infinitely many non-zero real values, there are infinitely many such polynomials.

Final Answer

The number of polynomials having zeroes −2 and 5 is infinite.

Conclusion

Hence, there are infinitely many polynomials whose zeroes are −2 and 5.