Prove Polynomial Big O

Show that $3x^2 + 25$ is on the order of $x^2$ or Big O of $x^2$.

In this problem, we are tasked with proving that a polynomial function, specifically three x squared plus twenty-five, is on the order of another polynomial, x squared. This exercise falls under the concept of asymptotic notation, which is crucial for understanding the efficiency of algorithms in computer science and mathematics. To solve problems like this, we need to demonstrate that the function grows at the same rate or slower than the function it is being compared to, which in this case is x squared. This is often done by showing that there exists a constant multiple of x squared that eventually dominates the function as x goes to infinity. This concept is represented mathematically as Big O notation and includes establishing bounds on a function's growth rate. The proof typically involves algebraic manipulations and sometimes calculus if the function's behavior needs more detailed analysis. This specific instance is straightforward since both functions are polynomials of the same degree, so the dominance is more intuitive. Exploring this problem helps cement the understanding of algorithmic efficiency, a skill that is indispensable not only in academic settings but also in practical computing applications.