If I have a polynomial function f(x) and I want to find an approximate polynomial g(x). I can apply the Nth order Taylor Polynomial of f centered at some value a. So suppose I have f(x) = 5 - 6x + 20x^3 + 10x^5 and I want to find an approximate function, g(x), centered at 5. From what I gather, this means that after I create the Taylor Polynomial of 5 degrees, when I plug in 5 to both f(x) and g(x), they will be equal values. That is, f(5) = g(5). However, does this also mean that f(x) = g(x) for all values of x (not just the centered value) if one polynomial is approximated (Taylor polynomial) from another polynomial? (as opposed to a polynomial being approximated from a non-polynomial)? Note: I'm not considering infinite series in this question, only nth term Taylor polynomials.