MATH SOLVE

3 months ago

Q:
# Find a third-degree polynomial equation with rational coefficients that has roots -4 and 6 + i.

Accepted Solution

A:

Answer:x³ - 8x² - 11x + 148Step-by-step explanation:Given that x = 6 + i is a root then x = 6 - i is also a rootComplex roots occur as conjugate pairs.The factors are therefore (x - (6 + i)) and(x - (6 - i))Given x = - 4 is a root then (x + 4) is a factor The polynomial is the product of the factors, that isp(x) = (x + 4)(x - (6 + i))(x - (6 - i)) = (x + 4)(x - 6 - i)(x - 6 + i) = (x + 4)((x - 6)² - i²) = (x + 4)(x² - 12x + 36 + 1) = (x + 4)(x² - 12x + 37) ← distribute = x³ + 4x² - 12x² - 48x + 37x + 148 = x³ - 8x² - 11x + 148