GATE Overflow Test Series | Quantitative Aptitude | Shallow Test 1: 7

Question

The polynomial \( x^4 - ax^3 + bx^2 - cx + 4 \) when divided by \( x - 2 \) leaves a remainder of 5 and when divided by \( x + 2 \) leaves a remainder of 11. Then \( b^2 = \)____

• A. 25
• B. 36
• C. 9
• D. 4

Answer 1

Given:
1. \( P(2) = 2^4 - a(2^3) + b(2^2) - c(2) + 4 = 5 \)
\[ 16 - 8a + 4b - 2c + 4 = 5 \]
\[ 16 - 8a + 4b - 2c + 4 = 5 \]
\[ 20 - 8a + 4b - 2c = 5 \]
\[ -8a + 4b - 2c = -15 \quad \text{(1)} \]

2. \( P(-2) = (-2)^4 - a(-2)^3 + b(-2)^2 - c(-2) + 4 = 11 \)
\[ 16 + 8a + 4b + 2c + 4 = 11 \]
\[ 20 + 8a + 4b + 2c = 11 \]
\[ 8a + 4b + 2c = -9 \quad \text{(2)} \]

By solving these two equations:
From (1):
\[ -8a + 4b - 2c = -15 \quad \text{(3)} \]

From (2):
\[ 8a + 4b + 2c = -9 \quad \text{(4)} \]

Adding (3) and (4):
\[ -8a + 4b - 2c + 8a + 4b + 2c = -15 - 9 \]
\[ 8b = -24 \]
\[ b = -3 \]

Finding \( b^2 \):
\[ b^2 = (-3)^2 = 9 \]

Correct Answer: C. 9