Question

Given a polynomial that has zeros of −3, 5i, and −5i and has a value of 816 when x=3. write the polynomial in standard form axn bxn−1 …. answer using reduced fractions when necessary.

Answer 1

To find the polynomial with the given zeros, we use the fact that complex zeros come in conjugate pairs. Finally, we multiply these factors: (x + 3)(x - 5i)(x + 5i) = (x + 3)((x - 5i)(x + 5i)) = (x + 3)(x² + 25) = x³ + 3x² + 25x + 75.

Step-by-step explanation:

To find the polynomial with the given zeros, we use the fact that complex zeros come in conjugate pairs. This means that if 5i is a zero, then -5i is also a zero. So, the factors of the polynomial are (x + 3)(x - 5i)(x + 5i).

To find the polynomial in standard form, we multiply these factors:

(x + 3)(x - 5i)(x + 5i) = (x + 3)((x - 5i)(x + 5i))

(x + 3)(x² - (5i)²) = (x + 3)(x² - 25i²)

(x + 3)(x² + 25) = x³ + 3x² + 25x + 75.