Question

find a third degree polynomial function of the lowest degree that has the zeros below and whose leading coefficient is one. 1, -5 -√2, -5+√2

Answer 1

They told you the zeroes. That means the polynomial factors into:
(x-1)(x-(-5-sqrt2))(x-(-5+sqrt2))
If you multiply this out, it should give you an equation of a third degree polynomial. Third degree means it will have an x^3 in it. Nothing bigger than x^3. So no x^4 or x^5....

Answer 2

Simplifying a bit:
(x-1)(x + 5 + sqrt2)(x + 5 - sqrt2)
If you multiply the right two brackets first, it's easier. You can simplify and the terms with radicals disappear.
Then multiply that result by (x-1)
The polynomial you end up with is: x^3 + 9x^2 + 13 -23