Question

Build a polynomial P with integer coefficients such that x = -2 - i is a zero, and P has a local maximum.

Answer 1

To build a polynomial P with integer coefficients such that x = -2 - i is a zero and P has a local maximum, we can use the conjugate theorem and the zero product property to construct the polynomial P(x) = x² + 4x + 5.

Step-by-step explanation:

To build a polynomial P with integer coefficients such that x = -2 - i is a zero and P has a local maximum, we can use the conjugate theorem. Since -2 - i is a zero, its conjugate -2 + i is also a zero.

Using the zero product property, we can write the polynomial as P(x) = (x - (-2 - i))(x - (-2 + i))

Expanding this, we get P(x) = (x + 2 + i)(x + 2 - i)

Simplifying further, we get P(x) = x² + 4x + 5.