1.) Find all zeros of the function f(x)=6x3−7x2−14x+8. and 2.) Given y=3x5−x4+9x3−3x2−12x+4, and that 2i is a zero, write y in factored form (as a product of linear factors). Be sure to write the full equation

1. To find the zeros of f(x)=6x^3−7x^2−14x+8, apply the Rational Root Theorem and synthetic division:
- Test potential rational roots: \pm 1, \pm 2, \pm 4, \pm 8, \pm \frac{1}{2}, \pm \frac{1}{3}, \pm \frac{2}{3}, \pm \frac{4}{3}, \pm \frac{8}{3}.
- Using synthetic division, we find that f\left(\frac{-2}{3}\right) = 0 and f(2) = 0.
- Further factorization shows:
(x + \frac{2}{3})(x - 2)(6x + 4) = 0
- Simplify to find the zeros:
x = -\frac{2}{3}, 2

Answer: x = -\frac{2}{3}, 2

2. To write y=3x^5−x^4+9x^3−3x^2−12x+4 in factored form given that 2i is a zero:
- Since 2i is a zero, -2i is also a zero due to complex conjugates appearing together.
- Using synthetic division, check additional real roots.
- Suppose 1, 2, -1 are real zeros (use synthetic division to verify).
- Factor together:
3(x - 2i)(x + 2i)(x - 1)(x - 2)(x + 1)

Answer: y = 3(x - 2i)(x + 2i)(x - 1)(x - 2)(x + 1)