To find the solutions for the equation x6 + 2x – 60 = 0, we can start by evaluating potential rational roots using the Rational Root Theorem or through numerical methods.
First, let’s rearrange the equation to look for rational roots:
x6 + 2x – 60 = 0
Next, we can try substituting some integer values for x to see if we can find any solutions. For example:
- If x = 2:
26 + 2(2) – 60 = 64 + 4 – 60 = 8 (not a solution) - If x = 3:
36 + 2(3) – 60 = 729 + 6 – 60 = 675 (not a solution) - If x = 1:
16 + 2(1) – 60 = 1 + 2 – 60 = -57 (not a solution) - If x = -2:
(-2)6 + 2(-2) – 60 = 64 – 4 – 60 = 0 (found a solution)
Thus, one of the solutions to the equation is x = -2.
However, finding all solutions may require more advanced methods, such as synthetic division or numerical root-finding algorithms. In this case, x = -2 is one solution, but you may explore others using polynomial solving techniques or graphing methods for a complete set of roots.