Final answer:
The polynomial 4x² + 9 factors into (2x + 3i)(2x - 3i), which gives the complex zeros -1.5i and 1.5i, each with a multiplicity of 1.
Step-by-step explanation:
To factor the polynomial P(x) = 4x² + 9 completely and find all its zeros, we first notice that this polynomial is not in the form of a quadratic ax² + bx + c = 0 where b and c are non-zero. Here, we have a polynomial without a linear term, and the constant term is positive, which means it cannot be factored into real number factors. Instead, we can use the difference of squares to factor it, since 4x² is a perfect square and so is 9. The factored form will look like this:
(2x + 3i)(2x - 3i) = 0
From there, we set each factor equal to zero to find the zeros:
- 2x + 3i = 0 → x = -½ * 3i
- 2x - 3i = 0 → x = ½ * 3i
The zeros of the polynomial are -1.5i and 1.5i with each zero having a multiplicity of 1.