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We are given the polynomial \( A(x) = x^6 + 5x^5 - 3x^4 + 9x^2 - 2x - 1 \), and we need to find two other polynomials \( B(x) \) and \( C(x) \) such that:

1. \( A(x) - B(x) = x^2 - 1 \)
2. \( A(x) - C(x) = x^3 + 2x - 6 \)

### Part a) \( A(x) - B(x) = x^2 - 1 \)
From the equation, we can express \( B(x) \) as:
\[
B(x) = A(x) - (x^2 - 1)
\]
Substitute \( A(x) \) into this:
\[
B(x) = (x^6 + 5x^5 - 3x^4 + 9x^2 - 2x - 1) - (x^2 - 1)
\]
Simplify:
\[
B(x) = x^6 + 5x^5 - 3x^4 + 9x^2 - 2x - 1 - x^2 + 1
\]
\[
B(x) = x^6 + 5x^5 - 3x^4 + 8x^2 - 2x
\]

### Part b) \( A(x) - C(x) = x^3 + 2x - 6 \)
Similarly, we can express \( C(x) \) as:
\[
C(x) = A(x) - (x^3 + 2x - 6)
\]
Substitute \( A(x) \):
\[
C(x) = (x^6 + 5x^5 - 3x^4 + 9x^2 - 2x - 1) - (x^3 + 2x - 6)
\]
Simplify:
\[
C(x) = x^6 + 5x^5 - 3x^4 - x^3 + 9x^2 - 4x + 5
\]

### Final Answers:
- \( B(x) = x^6 + 5x^5 - 3x^4 + 8x^2 - 2x \)
- \( C(x) = x^6 + 5x^5 - 3x^4 - x^3 + 9x^2 - 4x + 5 \)

Would you like more details or have any questions? Here are some related questions:

1. How do you subtract polynomials with higher degrees?
2. What is the general method to solve for unknown polynomials?
3. How do you handle polynomials when there are multiple variables involved?
4. Can we verify the solutions by plugging them back into the original equations?
5. How would the solution change if the degrees of the polynomials were different?

**Tip:** Always combine like terms carefully when subtracting or adding polynomials to avoid mistakes.

Cho đa thức: A(x) = x^6 + 5x^5 - 3x^4 + 9x^2 - 2x - 1. Tìm các đa thức B(x), C(x) sao cho: a) A(x) - B(x) = x^2 - 1. b) A(x) - C(x) = x^3 + 2x - 6.

Learn how to solve for unknown polynomials B(x) and C(x) by subtracting from the given polynomial A(x) = x^6 + 5x^5 - 3x^4 + 9x^2 - 2x - 1. This problem covers algebraic concepts and polynomial subtraction suitable for Grades 10-12.

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