if t=x^2+1/x^2, determine the values of b and c so that x^6+1/x^6=T^3+bT+ c for all non-zero real numbers x 

Given \( t = x^2 + \frac{1}{x^2} \), we need to determine the values of \( b \) and \( c \) such that:

\[ x^6 + \frac{1}{x^6} = t^3 + bt + c \]

First, we need to express \( x^6 + \frac{1}{x^6} \) in terms of \( t \).

1. **Express \( x^6 + \frac{1}{x^6} \) in terms of \( x^2 + \frac{1}{x^2} \)**:
   \[
   t = x^2 + \frac{1}{x^2}
   \]

2. **Square \( t \)**:
   \[
   t^2 = \left( x^2 + \frac{1}{x^2} \right)^2
   \]
   \[
   t^2 = x^4 + 2 + \frac{1}{x^4}
   \]
   Therefore,
   \[
   x^4 + \frac{1}{x^4} = t^2 - 2
   \]

3. **Square \( x^4 + \frac{1}{x^4} \)**:
   \[
   \left( x^4 + \frac{1}{x^4} \right)^2 = \left( t^2 - 2 \right)^2
   \]
   \[
   x^8 + 2 + \frac{1}{x^8} = t^4 - 4t^2 + 4
   \]

4. **Subtract 2 from both sides**:
   \[
   x^8 + \frac{1}{x^8} = t^4 - 4t^2 + 2
   \]

5. **Find \( x^6 + \frac{1}{x^6} \)** using the intermediate results**:
   Using the identity:
   \[
   (x^2 + \frac{1}{x^2})(x^4 + \frac{1}{x^4}) = x^6 + \frac{1}{x^6} + x^2 \cdot \frac{1}{x^4} + x^4 \cdot \frac{1}{x^2}
   \]
   \[
   t(t^2 - 2) = x^6 + \frac{1}{x^6} + t
   \]
   \[
   t^3 - 2t = x^6 + \frac{1}{x^6} + t
   \]
   \[
   x^6 + \frac{1}{x^6} = t^3 - 2t - t
   \]
   \[
   x^6 + \frac{1}{x^6} = t^3 - 3t
   \]

6. **Compare with the given equation**:
   \[
   x^6 + \frac{1}{x^6} = t^3 + bt + c
   \]
   By comparing the two equations:
   \[
   t^3 - 3t = t^3 + bt + c
   \]

7. **Equate the coefficients**:
   \[
   b = -3
   \]
   \[
   c = 0
   \]

Therefore, the values of \( b \) and \( c \) are:

\[ b = -3 \]
\[ c = 0 \]

Would you like more details or have any questions about this solution?

Here are some more questions you might consider:
1. How can you generalize the process for other powers of \( x \)?
2. What are some common identities involving powers of \( x \) and \( \frac{1}{x} \)?
3. Can you derive similar expressions for \( x^{12} + \frac{1}{x^{12}} \)?
4. How do you determine the coefficients in polynomial equations?
5. What is the significance of comparing coefficients in polynomial identities?

**Tip:** When dealing with polynomial identities, breaking down the expressions into intermediate steps and using known identities can greatly simplify the process of finding unknown coefficients.

Learn how to determine the values of b and c such that x^6 + 1/x^6 = t^3 + bt + c for all non-zero real numbers x, using the identity involving powers of x and 1/x. This problem involves advanced algebraic manipulation and is suitable for advanced high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation