Polynomial Write

Question

write a polynomial f(x) that satisfies the given conditions. Degree three polynomial with integer coefficents with zeros 3i and 8/5

Answer 1

If 3i is a zero, then its conjugate (-3i) will also be a zero.

When 3i is a zero, then (x - 3i) is a factor.

When -3i is a zero, then (x - -3i) or (x + 3i) is a factor.

When 8/5 is a zero, then (x - 8/5) is a factor. Since we are looking for integer coefficients, we can write the factor as (5x - 8). Explanation: If x - 8/5 is a factor, then we solve for the zero by setting the factor equal to zero meaning x - 8/5 = 0 or x = 8/5. We get the same result when we set 5x - 8 equal to zero: 5x - 8 = 0 means 5x = 8 or x = 8/5.

The polynomial then is:

P(x) = (x - 3i)(x + 3i)(5x - 8)

P(x) = (x² - 9i²)(5x - 8)

P(x) = (x² + 9)(5x - 8)

P(x) = 5x³ - 8x² + 45x - 72