To find the value of \( k \) that satisfies a specific condition for the polynomial \( p(x) = x^6 + kx^4 - x^3 + 3x \), we need to define the condition. Here are some possible scenarios for determining \( k \):

1. **Given a Root**:
   If we know that \( x = a \) is a root of the polynomial, then:
   \[
   p(a) = a^6 + ka^4 - a^3 + 3a = 0
   \]
   Solving this equation will give us \( k \).

2. **Given a Critical Point**:
   If we know that \( x = b \) is a critical point, then:
   \[
   p'(b) = 6b^5 + 4kb^3 - 3b^2 + 3 = 0
   \]
   Solving this equation for \( k \) will give us the required value.

3. **Specific Value of \( p(x) \) at a Point**:
   If we know the value of the polynomial at a specific point, say \( p(c) = d \), then:
   \[
   p(c) = c^6 + kc^4 - c^3 + 3c = d
   \]
   Solving this equation will give us \( k \).

### Example Problem

Suppose we are given that \( x = 1 \) is a root of the polynomial. We substitute \( x = 1 \) into the polynomial and set it equal to zero to find \( k \):

\[
p(1) = 1^6 + k \cdot 1^4 - 1^3 + 3 \cdot 1 = 0
\]

This simplifies to:

\[
1 + k - 1 + 3 = 0
\]

\[
k + 3 = 0
\]

\[
k = -3
\]

So, if \( x = 1 \) is a root of the polynomial, then \( k = -3 \).

### Another Example

Suppose we are given that \( x = 2 \) is a critical point. We substitute \( x = 2 \) into the first derivative and set it equal to zero to find \( k \):

\[
p'(2) = 6 \cdot 2^5 + 4k \cdot 2^3 - 3 \cdot 2^2 + 3 = 0
\]

This simplifies to:

\[
6 \cdot 32 + 4k \cdot 8 - 3 \cdot 4 + 3 = 0
\]

\[
192 + 32k - 12 + 3 = 0
\]

\[
183 + 32k = 0
\]

\[
32k = -183
\]

\[
k = -\frac{183}{32}
\]

### Summary

The value of \( k \) depends on the specific conditions given for the polynomial. By substituting the known conditions into the polynomial or its derivatives, we can solve for \( k \).

### Questions and Exploration

1. What are the roots of the polynomial when \( k = 2 \)?
2. How does the polynomial behave for large positive and negative values of \( x \)?
3. Can we find \( k \) if we know \( p'(x) \) has a specific value at a given point?
4. What is the effect of changing \( k \) on the symmetry of the polynomial?
5. How many real roots can the polynomial have for different values of \( k \)?
6. How does the polynomial's graph change as \( k \) varies from negative to positive?
7. What are the inflection points for a specific \( k \)?
8. Can we use the quadratic formula to solve for \( k \) in any of the equations derived from \( p(x) \) or \( p'(x) \)?

### Tip

When dealing with polynomials, always verify the conditions given (roots, critical points, values) and substitute them carefully into the polynomial or its derivatives. This systematic approach helps in accurately finding the unknown coefficients like \( k \).

Discover how to find the value of k in the polynomial p(x) = x^6 + kx^4 - x^3 + 3x under specific conditions such as roots and critical points. This problem involves evaluating the polynomial and its derivative to determine k. Suitable for advanced high school students and beyond.

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