The number of polynomial having zeroes as -2 and 5 is --------(A) 1 (B)2 (C) 3 (D) more than 3
more than 3.
f(x) = a(x+2)(x-5)
will have those two zeroes. But a can have any value you like.
Also, f may have higher degree than 2, since
a(x+2)(x-5)*p(x)
will also have those two roots, and p(x) can be any polynomial you like.
To find the number of polynomials that have zeroes at -2 and 5, we need to consider a few factors. First, we know that a polynomial of degree n will have at most n distinct zeroes.
In this case, we are given that the polynomial has zeroes at -2 and 5. Therefore, the polynomial must be of at least degree 2 (since it has two distinct zeroes).
To construct such a polynomial, we can use the fact that if a polynomial has a zero at a particular value, say c, then it must have a factor of (x - c) in its factored form. So, for a polynomial with zeroes at -2 and 5, we would have factors of (x - (-2)) and (x - 5), which simplifies to (x + 2) and (x - 5), respectively.
From these factors, we can construct different polynomials by multiplying them together and potentially adding other factors. For example:
- (x + 2) * (x - 5) = x^2 - 3x - 10
- (x + 2) * (x - 5) * 2 = 2x^2 - 6x - 20
Therefore, we can see that there are more than 3 polynomials that have zeroes at -2 and 5. Hence, the answer is (D) more than 3.