Problem 1. What is the long run behavior of the following polynomial function? (Or, how do the graphs of polynomial look like in long run?) f(x)=2x^2-x+1 b) f(x)=x^100+x^50-2 c) f(x)=3x^7-x^5+x

To determine the long-run behavior of polynomial functions, we need to consider the leading term, which is the term with the highest power of x.<br /><br />a) For the polynomial function $f(x)=2x^{2}-x+1$, the leading term is $2x^{2}$. Since the coefficient of the leading term is positive, the graph of the polynomial will rise to positive infinity as x approaches positive infinity and will rise to positive infinity as x approaches negative infinity.<br /><br />b) For the polynomial function $f(x)=x^{100}+x^{50}-2$, the leading term is $x^{100}$. Since the coefficient of the leading term is positive, the graph of the polynomial will rise to positive infinity as x approaches positive infinity and will rise to positive infinity as x approaches negative infinity.<br /><br />c) For the polynomial function $f(x)=3x^{7}-x^{5}+x$, the leading term is $3x^{7}$. Since the coefficient of the leading term is positive, the graph of the polynomial will rise to positive infinity as x approaches positive infinity and will fall to negative infinity as x approaches negative infinity.<br /><br />In summary, the long-run behavior of polynomial functions depends on the sign of the coefficient of the leading term. If the coefficient is positive, the graph will rise to positive infinity as x approaches positive infinity and will rise to positive infinity as x approaches negative infinity. If the coefficient is negative, the graph will fall to negative infinity as x approaches positive infinity and will fall to negative infinity as x approaches negative infinity.