Question

If \(x+1\) is a factor of \(2x^3 + ax^2 + 2bx + 1\), then find the values of \(a\) and \(b\) given that \(2a - 3b = 4\).

• A. \(5, 3\)
• B. \(5, 2\)
• C. \(3, 4\)
• D. \(2, 7\)
• E. \(3, 6\)

Answer 1

The Correct Option is B

Concept: Using
Factor Theorem:

• If \(x+1\) is a factor, then \(f(-1) = 0\)

Step 1: Apply factor theorem.
\[ f(x) = 2x^3 + ax^2 + 2bx + 1 \] \[ f(-1) = 2(-1)^3 + a(-1)^2 + 2b(-1) + 1 = 0 \] \[ -2 + a - 2b + 1 = 0 \Rightarrow a - 2b - 1 = 0 \Rightarrow a = 2b + 1 \quad \cdots (1) \]
Step 2: Use given condition.
\[ 2a - 3b = 4 \quad \cdots (2) \]
Step 3: Substitute (1) into (2).
\[ 2(2b+1) - 3b = 4 \] \[ 4b + 2 - 3b = 4 \Rightarrow b + 2 = 4 \Rightarrow b = 2 \]
Step 4: Find \(a\).
\[ a = 2b + 1 = 2(2) + 1 = 5 \]
Step 5: Option analysis.

• (A) (5,3): Does not satisfy equation $\times$
• (B) (5,2): Correct \checkmark
• (C) (3,4): Incorrect $\times$
• (D) (2,7): Incorrect $\times$
• (E) (3,6): Incorrect $\times$

Conclusion:
Thus, the correct answer is
Option (B).