Express the following equation, using O ? or T notation

Question

Express the following equation, using Ω Ο or Θ notation
|3x^5 -4x^4 +3x^3 +2x^2 +5x +12| ≤ 12|x^5 |

Answer 1

This says that f(n) =3x^5-4x^4+3x^3+2x^2+5x+12 =O (x^5) for  constant of 12.

To verify thisdivide both sidby of the original equation by x^5:

3-4/x+3/x^2+2/x^3+5/x^4+12/x^5<=12 as x approaches infinity.

So since all the terms divided by x or some power of x go to zero, this simplifies to 3 <= 12. Check!