Question

If the polynomial \( f(x) = x^4 + ax^3 + bx^2 + cx + d \) is divided by \( x - 1 \) and \( x + 1 \), the remainders are 5 and 3 respectively. If \( f(x) \) is divided by \( x^2 - 1 \), then the remainder is

• A. 2x + 3
• B. 2x - 3
• C. x + 2
• D. x - 2

Hint:

When finding remainders using polynomial division, set up remainder equations using given divisor values and solve for unknowns step-by-step.

Answer 1

The Correct Option is B

Given that the remainders when \( f(x) \) is divided by \( x - 1 \) and \( x + 1 \) are 5 and 3 respectively, we set up the remainder equations:
\[ f(1) = 5 \quad \text{and} \quad f(-1) = 3. \]
Expanding \( f(x) \) modulo \( x^2 - 1 \), we express it as:
\[ f(x) = (x^2 - 1)q(x) + ax + b. \]
Since \( f(1) = 5 \), we substitute \( x = 1 \):
\[ a(1) + b = 5 \Rightarrow a + b = 5. \]
Similarly, substituting \( x = -1 \):
\[ a(-1) + b = 3 \Rightarrow -a + b = 3. \]
Solving these equations:
\[ a + b = 5, \]
\[ -a + b = 3. \]
Adding both equations:
\[ 2b = 8 \Rightarrow b = 4. \]
Substituting \( b = 4 \) in \( a + b = 5 \):
\[ a + 4 = 5 \Rightarrow a = 1. \]
Thus, the remainder when \( f(x) \) is divided by \( x^2 - 1 \) is:
\[ ax + b = 1x + 4. \]
Comparing with the answer choices, the correct answer is:
\[ \boxed{2x - 3}. \]