Determine the roots of 9x³-18x²-24x-8
To determine the roots of the polynomial 9x³ - 18x² - 24x - 8, we can start by factoring out the greatest common factor, which is 1:
1(9x³ - 18x² - 24x - 8)
Next, we can check for any potential rational roots using the rational root theorem. The potential rational roots of the polynomial are determined by dividing the factors of the constant term (-8) by the factors of the leading coefficient (9).
The factors of 9 are ±1, ±3, and ±9.
The factors of 8 are ±1, ±2, ±4, ±8.
Therefore, the potential rational roots are: ±1/1, ±2/1, ±4/1, ±8/1, ±1/3, ±2/3, ±3/3, ±8/3, ±1/9, ±2/9, ±3/9, and ±8/9.
Next, we can use synthetic division or long division to check each potential root. After checking all the potential roots, we find that the only real root is x = 2/3.
Therefore, the polynomial 9x³ - 18x² - 24x - 8 can be factored as (x - 2/3)(9x² - 12x - 12).
To find the remaining complex roots, we can solve the quadratic equation 9x² - 12x - 12 = 0 using the quadratic formula:
x = (-(-12) ± √((-12)² - 4(9)(-12))) / (2(9))
x = (12 ± √(144 + 432)) / 18
x = (12 ± √576) / 18
x = (12 ± 24) / 18
This gives us two complex roots:
x₁ = (12 + 24) / 18 = 36 / 18 = 2
x₂ = (12 - 24) / 18 = -12 / 18 = -2/3
Therefore, the roots of 9x³ - 18x² - 24x - 8 are x = 2/3 (real) and x = 2, -2/3 (complex).