To analyze and solve the given polynomial function $p(x) = x^6 + kx^4 - x^3 + 3x$, let's break it down step-by-step.

Polynomial Function Analysis

1. Degree of the Polynomial:

   • The highest power of $x$ in the polynomial $p(x)$ is $x^6$, so it is a 6th-degree polynomial.

2. Leading Coefficient:

   • The leading coefficient (the coefficient of the term with the highest power) is 1.

3. Coefficients:

   • The coefficients of the polynomial are: $1$ (for $x^6$), $k$ (for $x^4$), $-1$ (for $x^3$), and $3$ (for $x$).

4. Behavior at Infinity:

   • As $x \to \infty$ or $x \to -\infty$, the term $x^6$ will dominate, and since its coefficient is positive, $p(x) \to \infty$.

5. Roots of the Polynomial:

   • To find the roots of the polynomial, we need to solve $p(x) = 0$: $x^6 + kx^4 - x^3 + 3x = 0$
   • This might require numerical methods or specific techniques depending on the value of $k$.

Steps to Find Critical Points

To find critical points (where the first derivative $p'(x)$ is zero or undefined):

1. First Derivative: $p'(x) = 6x^5 + 4kx^3 - 3x^2 + 3$

2. Set the First Derivative to Zero: $6x^5 + 4kx^3 - 3x^2 + 3 = 0$

3. Solve for $x$:

   • This may involve factoring or using numerical methods.

Example with a Specific $k$

If $k = 2$, the polynomial becomes: $p(x) = x^6 + 2x^4 - x^3 + 3x$

1. First Derivative: $p'(x) = 6x^5 + 8x^3 - 3x^2 + 3$

2. Set the First Derivative to Zero: $6x^5 + 8x^3 - 3x^2 + 3 = 0$

3. Solve for $x$:

   • This might be solved using numerical methods like Newton's method or approximations.

Further Analysis

For a detailed analysis, more steps would involve:

• Second Derivative: $p''(x) = 30x^4 + 24kx^2 - 6x$

• Inflection Points:

  • Set the second derivative to zero and solve for $x$: $30x^4 + 24kx^2 - 6x = 0$

Questions and Exploration

1. What happens to the polynomial if $k$ is a negative value?
2. How does the shape of the graph change with different values of $k$?
3. What are the critical points for a specific $k$?
4. How do we find the roots of the polynomial numerically?
5. Can we determine if the polynomial has any local maxima or minima?
6. How can we approximate the roots using Newton's method?
7. What is the behavior of the polynomial around $x = 0$?
8. How does the term $-x^3$ influence the polynomial's behavior?

Tip

When analyzing polynomials, always start by examining the degree, leading coefficient, and behavior at infinity, as these provide insights into the general shape of the graph.