Find a third-degree polynomial equation with rational coefficients that has roots –4 and 6 + i. (1 point) Responses x3 – 8x2 – 11x + 148 = 0 x 3 – 8 x 2 – 11 x + 148 = 0 x3 – 8x2 – 12x + 37 = 0 x 3 – 8 x 2 – 12 x + 37 = 0 x3 – 12x2 + 37x = 0 x 3 – 12 x 2 + 37 x = 0 x3 – 8x2 – 11x = 0 x 3 – 8 x 2 – 11 x = 0
The correct polynomial equation is:
(x + 4)(x - 6 - i)(x - 6 + i) = 0
Expanding this equation, we get:
(x + 4)((x - 6) - i)((x - 6) + i) = 0
(x + 4)((x - 6)² - i²) = 0
(x + 4)((x - 6)² + 1) = 0
(x + 4)(x² - 12x + 37) = 0
Expanding this equation further, we get:
x³ - 8x² - 11x + 148 = 0
Therefore, the correct third-degree polynomial equation with rational coefficients that has roots –4 and 6 + i is:
x³ - 8x² - 11x + 148 = 0.