# Now, suppose one of the roots of the polynomial function is complex.

What is the equation for the polynomial function with roots 2 + i and 5? And which of the following must also be a root of the function: −3, −5, 2 − i, 2i?

The equation of the polynomial function is x³ - 9x² + 25x - 25 = 0. The other root of the polynomial is 2 - i. Given that the roots are 2 + i and 5, the equation for the polynomial function is x³ - 9x² + 25x - 25 = 0. Therefore, the roots of the function are 2 + i, 5, and 2 - i. This means that 2 - i must also be a root of the function.

## The Complex Conjugate Theorem

**Equation for the Polynomial Function**To find the equation for the polynomial function with roots 2 + i and 5, we can use the factored form of the polynomial. The roots of the function are 5, 2 + i, and 2 - i. Therefore, the equation can be expressed as (x - 5)(x - (2 + i))(x - (2 - i)) = 0. By expanding the expression, we get: (x - 5)(x - 2 - i)(x - 2 + i) = 0 (x - 5)((x - 2)² - i²) = 0 (x - 5)(x² - 4x + 5) = 0 x(x² - 4x + 5) - 5(x² - 4x + 5) = 0 x³ - 9x² + 25x - 25 = 0 Therefore, the equation for the polynomial function with roots 2 + i and 5 is x³ - 9x² + 25x - 25 = 0.