How do you find the degree, leading term, the leading coefficient, the constant term and the end behavior of #g(x)=3x^5-2x^2+x+1#?

Question

Cora Alexander · Accepted Answer

Degree: #8#
Leading term: #3x^5#
Leading Coefficient: #3#
Constant: #1#
End behavior: See below in blue

The degree is the sum of the exponents on all terms. Our exponents are #5, 2# and #1#, which sum up to #8#. This is the degree of our polynomial #g(x)#. The leading term of a polynomial is just the term with the highest degree, and we see this is #3x^5#. The leading coefficient is just the number multiplying the highest degree term. The coefficient on #3x^5# is #3#. The constant term is just a term without a variable. In our case, the constant is #1#. For end behavior, we want to consider what our function goes to as #x# approaches positive and negative infinity. In our polynomial #g(x)#, the term with the highest degree is what will dominate the end behavior. So let's take the limit of it: #color(blue)(lim_(xrarroo) 3x^5=oo)# #color(blue)(lim_(xrarr-oo) 3x^5=-oo)# Our limits describe our limit behavior. Hope this helps!

Theodore Amidon · Answer

undefined Degree: 5
Leading term: 3x^5
Leading coefficient: 3
Constant term: 1
End behavior: As x approaches positive or negative infinity, g(x) approaches positive infinity.