Let's analyze the expression given: $f(x) = (2 - x)^7$.

1. Degree:

The degree of a polynomial is the highest power of $x$ in the expression when expanded. Since $(2 - x)^7$ has an exponent of 7, the degree of this polynomial is: $\text{Degree} = 7$

2. Leading Coefficient:

To find the leading coefficient, expand the term $(2 - x)^7$ and observe the coefficient of the term with the highest power of $x$, which is $x^7$. The expansion will produce a term with $-x$ raised to the 7th power, resulting in $-x^7$. Therefore, the leading coefficient is: $\text{Leading Coefficient} = -1$

3. Limits:

For the limits at $x \to -\infty$ and $x \to \infty$, consider the behavior of $f(x) = (2 - x)^7$ as $x$ approaches these values.

• As $x \to \infty$: $f(x) = (2 - x)^7 \approx (-x)^7 = -x^7 \to -\infty$

• As $x \to -\infty$: $f(x) = (2 - x)^7 \approx (-x)^7 = -x^7 \to \infty$

Summary:

• Degree: 7
• Leading Coefficient: -1
• $\lim_{x \to \infty} f(x) = -\infty$
• $\lim_{x \to -\infty} f(x) = \infty$

Would you like further details on any specific part?

Here are some related questions to expand your understanding:

1. How does the sign of the leading coefficient affect the end behavior of a polynomial?
2. What is the general method for finding the degree and leading coefficient of a polynomial expression?
3. How would the limits change if the power of $x$ was even instead of odd?
4. Can the behavior of limits at infinity differ if we change the base term to a positive expression, like $(x + 2)^7$?
5. How can the Binomial Theorem help in expanding expressions like $(2 - x)^7$?

Tip: For polynomial functions, the degree and leading coefficient provide key insights into the function’s growth rate and direction at infinity.