A polynomial P(x) has some number α as a zero if and only if x−α is a factor of P(x). To generate a polynomial with desired zeros, then, we can multiply any such factors.
As our desired polynomial has −2 as a zero, it must have a factor of x−(−2)=x+2. As no other specific zero is given, we can make that choice ourselves. Suppose the other zero (possibly also being −2), is k. Then the polynomial would be
P(x)=(x+2)(x−k)
=x2+(2−k)x−2k
Choosing any value for k will give a degree 2 polynomial with −2 as a zero. For example, k=0 gives x2+2x, or k=2 gives x2−4. Multiplying by any nonzero constant also will result in a valid polynomial.
