Yeah I usually use wolfram for finding zeros of high degree polynomials, but only after I forced myself to learn the rational root theorem. I am self teaching myself math so I'm not sure what I'll need what for, so I choose to learn it anyways(Havent started calculus yet, but my goal is to do econometrics). A method that, although time consuming, has proven fruitful in insight. My goal is to have a deep understanding of math, not just to "get by", as I suppose most students tend to do(I used to be one).
How is it relevant to a calculus course? In fact, I predict that within five to ten years one will hard pressed to find a calculus textbook that includes a chapter on techniques of integration. I have taught calculus on and off since 1964. I never thought that technology would replace basics techniques. But it has and will continue to do so. Partial fractions are gone as another victim of technology.