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#?

1 Answer
Jun 15, 2018

Answer:

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

Explanation:

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!