To solve this problem, we need to determine the degree of each polynomial function provided in the exercise. The degree of a polynomial is the highest power of the variable $x$ in the expression.

Let's go through each part:

1. (a) $f(x) = 6x^7 + 2x^5 - 4x + 1$

   • The highest power of $x$ is $7$, so the degree of $f(x)$ is 7.

2. (b) $g(x) = (x - 5)^3$

   • Expanding $(x - 5)^3$ would give a polynomial with the highest power of $x^3$, so the degree of $g(x)$ is 3.

3. (c) $k(x) = x(x - 2x^2)$

   • Simplify inside the parentheses: $x - 2x^2 = -2x^2 + x$.
   • Multiplying by $x$: $x \cdot (-2x^2 + x) = -2x^3 + x^2$.
   • The highest power of $x$ is $3$, so the degree of $k(x)$ is 3.

4. (d) $n(x) = x^2(x - 3)^2$

   • Expanding $(x - 3)^2 = x^2 - 6x + 9$.
   • Multiplying by $x^2$: $x^2 \cdot (x^2 - 6x + 9) = x^4 - 6x^3 + 9x^2$.
   • The highest power of $x$ is $4$, so the degree of $n(x)$ is 4.

Summary of Degrees:

• (a) Degree of $f(x)$: 7
• (b) Degree of $g(x)$: 3
• (c) Degree of $k(x)$: 3
• (d) Degree of $n(x)$: 4

Would you like further details or clarification on any part of this solution?

Additional Questions to Consider:

1. How would the degree change if we added a new term with a higher power to each polynomial?
2. What is the general rule for finding the degree of a polynomial that is a product of terms?
3. How does expanding expressions like $(x - 3)^2$ affect the degree?
4. Could we find the degree of these polynomials without expanding?
5. What is the difference between the degree of a polynomial and the leading coefficient?

Tip: Always look for the highest power of the variable in each term to quickly identify the degree of a polynomial.