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
The complex conjugate root theorem states that if a polynomial function with real coefficients has a complex root a + ib, where a and b are real numbers, then its complex conjugate a - ib is also a root of the function. In this case, if 2 + i is a root of the polynomial function, then 2 - i must also be a root.
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.