To determine if any of the given algebraic expressions \( A \), \( B \), \( C \), and \( D \) are equivalent, we need to simplify and compare them. Here are the expressions again for clarity:
- \( A = 2x^3 - x^2 - 6x \)
- \( B = 2x^3 + 8x + 4 \)
- \( C = 3x^4 + x^2 + x - 7 \)
- \( D = 3x^4 - 3x^2 + 5x - 7 \)
The task is to check if any of these expressions are equivalent to each other. Two expressions are equivalent if they simplify to the same polynomial form.
Let's consider this step-by-step:
1. Inspect the Structure: Visually inspect the degree and coefficients of each term in the polynomials.
- \( A \) and \( B \) are cubic polynomials (degree 3): highest term is \( x^3 \).
- \( C \) and \( D \) are quartic polynomials (degree 4): highest term is \( x^4 \).
2. Compare Equivalent Structures:
- For \( A \) and \( B \):
- Compare the coefficients of like terms (e.g., \( x^3 \), \( x^2 \), \( x \), and constant terms).
- \( A \) and \( B \): The coefficients for \( x^3 \) are the same (2). However, coefficients for \( x^2 \) are different (-1 for \( A \) and 0 for \( B \)), as well as for \( x \) and the constant terms.
- Hence, \( A \) and \( B \) are not equivalent.
- For \( C \) and \( D \):
- Compare the coefficients of like terms.
- \( C \) and \( D \): The coefficients for \( x^4 \) are the same (3). However, coefficients for \( x^2 \), \( x \), and constant terms are different.
- Hence, \( C \) and \( D \) are not equivalent.
3. Cross-Compare Different Degree Polynomials:
- \( A \) and \( C \), \( A \) and \( D \), \( B \) and \( C \), \( B \) and \( D \):
- Comparing polynomials of different degrees shows they cannot be equivalent since their highest degree terms are different (e.g., cubic vs. quartic).
After thorough inspection, none of the expressions simplify to the same polynomial form.
Based on the detailed comparison, there are no pairs of equivalent expressions among \( A \), \( B \), \( C \), and \( D \).
So, the answer is:
No two expressions from A, B, C, and D are equivalent.
