Question

The equations \( x^5 + ax + 1 = 0 \) and \( x^6 + ax^2 + 1 = 0 \) have a common root. Then \( a \) is equal to:

• A. \( -4 \)
• B. \( -2 \)
• C. \( -3 \)
• D. \( -1 \)
• E. \( 0 \)

Hint:

To find common roots, try to make the highest degree terms equal in both equations and then subtract them. This often reduces the problem to a much lower degree equation.

Answer 1

The Correct Option is B

Concept: If two equations have a common root, say \( \alpha \), then \( \alpha \) must satisfy both equations simultaneously. We can manipulate the two equations (usually by subtraction or substitution) to eliminate higher powers and solve for the coefficient \( a \).

Step 1: Manipulating the equations.
Let \( \alpha \) be the common root: \[ 1) \quad \alpha^5 + a\alpha + 1 = 0 \] \[ 2) \quad \alpha^6 + a\alpha^2 + 1 = 0 \] Multiply equation (1) by \( \alpha \): \[ 3) \quad \alpha^6 + a\alpha^2 + \alpha = 0 \] Subtract equation (2) from (3): \[ (\alpha^6 + a\alpha^2 + \alpha) - (\alpha^6 + a\alpha^2 + 1) = 0 \implies \alpha - 1 = 0 \implies \alpha = 1 \]

Step 2: Solving for \( a \).
Substitute the common root \( \alpha = 1 \) into equation (1): \[ (1)^5 + a(1) + 1 = 0 \] \[ 1 + a + 1 = 0 \implies a = -2 \]