Question

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`?

Answer 1

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!