Step-by-step explanation:
To check if a polynomial \( f(a) = 2a^4 + 5a^3 - 14a^2 - 5a + 15 \) gives the same remainder when divided by \( a-1 \) and \( a+1 \), we can apply the Remainder Theorem.
### Remainder when divided by \( a-1 \):
- Substitute \( a = 1 \) into \( f(a) \) to find \( f(1) \):
\[ f(1) = 2(1)^4 + 5(1)^3 - 14(1)^2 - 5(1) + 15 \]
\[ f(1) = 2 + 5 - 14 - 5 + 15 \]
\[ f(1) = 3 \]
So, the remainder when \( f(a) \) is divided by \( a-1 \) is \( 3 \).
### Remainder when divided by \( a+1 \):
- Substitute \( a = -1 \) into \( f(a) \) to find \( f(-1) \):
\[ f(-1) = 2(-1)^4 + 5(-1)^3 - 14(-1)^2 - 5(-1) + 15 \]
\[ f(-1) = 2 - 5 - 14 + 5 + 15 \]
\[ f(-1) = 3 \]
So, the remainder when \( f(a) \) is divided by \( a+1 \) is also \( 3 \).
### Conclusion:
Since \( f(1) = f(-1) = 3 \), we observe that the polynomial \( f(a) = 2a^4 + 5a^3 - 14a^2 - 5a + 15 \) gives the same remainder \( 3 \) when divided by both \( a-1 \) and \( a+1 \).
Therefore, it is indeed possible for a polynomial to give the same remainder when divided by divisors that differ by \( 2 \) (in this case, \( a-1 \) and \( a+1 \)). This aligns with the properties of polynomial division and the Remainder Theorem.
