One of the roots of 2x^3 − 9x^2 + 13x + k = 0 is x = 2. Find the other two roots.

Answer 1

To find the other two roots, we can use polynomial division or synthetic division to divide the given polynomial by (x - 2) since x = 2 is a root.

Using synthetic division:
2|2 -9 13 k
2 -4 18
___________
2 -7 9 k

Therefore, the polynomial can be factored as:
2x^3 − 9x^2 + 13x + k = (x - 2)(2x^2 - 7x + 9)

Now, to find the other two roots, we can solve the quadratic equation 2x^2 - 7x + 9 = 0 using the quadratic formula:
x = (-(-7) ± √((-7)^2 - 4*2*9)) / 2*2
x = (7 ± √(49 - 72)) / 4
x = (7 ± √(-23)) / 4
Since the discriminant is negative, the quadratic equation has two complex roots:
x = (7 ± √23i) / 4

Therefore, the other two roots are:
x = (7 + √23i) / 4
x = (7 - √23i) / 4