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

1 Answer
Aug 28, 2015

For even degree polynomials, a positive leading coefficient implies y+ as x±, while a negative leading coefficient implies y as x±. For odd degree polynomials, a positive leading coefficient implies y+ as x+ and y as x, while a negative leading coefficient implies y as x+ and y+ as x.

Explanation:

A (real) polynomial of (integer) degree n is a function of the form p(x)=anxn+an1xn1+an2xn2++a2x2+a1x+a0, where an0 (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 an>0 implies that y+ as x± and an<0 implies y as x±.

If n is odd, then an>0 implies that y+ as x+ and y as x and an<0 implies that y as x+ and y+ as x.

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