To analyze the given polynomial:

The function is: $f(x) = 2x^5 + 10x^4 - 6x^3 - 90x^2 - 108x$

Step 1: Identify the degree of the polynomial. The degree of a polynomial is the highest power of $x$ in the expression. In this case, the highest power is $x^5$.

Degree: $5$

Step 2: Identify the leading coefficient. The leading coefficient is the coefficient of the term with the highest power of $x$. Here, the term with $x^5$ has a coefficient of $2$.

Leading coefficient: $2$

Final Answer

The degree of the polynomial is 5, and the leading coefficient is 2.

Would you like more details or have any questions?

Here are 5 related questions:

1. What is the behavior of this polynomial as $x \to \infty$ and $x \to -\infty$?
2. How many roots can a 5th-degree polynomial have?
3. Can you identify any potential rational roots using the Rational Root Theorem?
4. What are the turning points of a polynomial of degree 5?
5. How would you factor a polynomial like this one?

Tip

Always start analyzing a polynomial by identifying the highest degree term and its coefficient, as these define many key properties of the function.