Question

which polynomial correctly combines the like terms and expresses the given polynomial in standard form?
8mn⁵ - 2m⁶ + 5m²n⁴ - m³n³ + n⁶ - 4m⁶ + 9m²n⁴ - mn⁵ - 4m³n³
○ n⁶ + 7mn⁵ + 14m²n⁴ - 5m³n³ - 6m⁶
○ -2m⁶ - 5m³n³ + 14m²n⁴ + 7mn⁵ + n⁶
○ 14m²n⁴ + 7mn⁵ - 6m⁶ - 5m³n³ + n⁶
○ n⁶ - 6m⁶ + 7mn⁵ + 14m²n⁴ - 5m³n³

Explanation:

Step1: Identify like terms

• For \( m^6 \): \( -2m^6 - 4m^6 \)
• For \( mn^5 \): \( 8mn^5 - mn^5 \)
• For \( m^2n^4 \): \( 5m^2n^4 + 9m^2n^4 \)
• For \( m^3n^3 \): \( -m^3n^3 - 4m^3n^3 \)
• For \( n^6 \): \( n^6 \)

Step2: Combine like terms

• \( m^6 \): \( -2m^6 - 4m^6 = -6m^6 \)
• \( mn^5 \): \( 8mn^5 - mn^5 = 7mn^5 \)
• \( m^2n^4 \): \( 5m^2n^4 + 9m^2n^4 = 14m^2n^4 \)
• \( m^3n^3 \): \( -m^3n^3 - 4m^3n^3 = -5m^3n^3 \)
• \( n^6 \): \( n^6 \)

Step3: Write in standard form (descending order of exponents, considering total degree)

Total degree of \( n^6 \) is 6, \( mn^5 \) is \( 1 + 5 = 6 \), \( m^2n^4 \) is \( 2 + 4 = 6 \), \( m^3n^3 \) is \( 3 + 3 = 6 \), \( m^6 \) is 6. Arrange terms (order can vary as long as like degrees are grouped, but let's check the options). The combined polynomial is \( n^6 + 7mn^5 + 14m^2n^4 - 5m^3n^3 - 6m^6 \) (or other equivalent orderings, but this matches the first option).

Answer:

\( n^6 + 7mn^5 + 14m^2n^4 - 5m^3n^3 - 6m^6 \) (the first option: \( n^6 + 7mn^5 + 14m^2n^4 - 5m^3n^3 - 6m^6 \))