How do the coefficients of a polynomial affects its end behavior?

1 Answer
Aug 28, 2015

Answer:

For even degree polynomials, a positive leading coefficient implies #y->+infty# as #x->pm infty#, while a negative leading coefficient implies #y->-infty# as #x->pm infty#. For odd degree polynomials, a positive leading coefficient implies #y->+infty# as #x->+ infty# and #y->-infty# as #x->-infty#, while a negative leading coefficient implies #y->-infty# as #x->+ infty# and #y->+infty# as #x->-infty#.

Explanation:

A (real) polynomial of (integer) degree #n# is a function of the form #p(x)=a_{n}x^{n}+a_{n-1}x^[n-1}+a_{n-2}x^{n-2}+cdots+a_{2}x^2+a_{1}x+a_{0}#, where #a_{n} != 0# (otherwise it wouldn't be degree #n#), and all the other #a#'s are arbitrary real numbers (and they can be zero).

If #n# is even, then #a_{n}>0# implies that #y->+infty# as #x->pm infty# and #a_{n}<0# implies #y->-infty# as #x->pm infty#.

If #n# is odd, then #a_{n}>0# implies that #y->+infty# as #x->+ infty# and #y->-infty# as #x->-infty# and #a_{n}<0# implies that #y->-infty# as #x->+ infty# and #y->+infty# as #x->-infty#.

The values of the other coefficients are irrelevant for determining the end behavior.