The sign and degree of the leading coefficient of a polynomial describes its end behavior.
"Degree" refers to the highest exponent of a variable. Polynomials of even degrees, like x2, x4, etc. have ends that go in the same direction. If the highest degree term is negative, both ends go down. If the highest degree term is positive, both ends go up. The easiest way to remember this is just to think of what x2 looks like.
Polynomials of odd degrees have ends that go opposite ways. Think about x3. Down on the left, up on the right. It's the same with a fifth degree polynomial, where the highest term is 4x5, x5, or (x5)/2. Ditto with a 7th, 9th, any odd degree polynomial. Down on the left, up on the right, unless the leading term is negative. Then, like -x3 it goes up on the left and down on the right.
***NOTE*** The leading coefficient refers to the first term when a polynomial is in standard form. Standard form is when the highest degree term is first, and the terms follow in descending order of degree, often ending with a constant.
Hope that helps!
