The value of a for which the equation `x^3 + ax + 1 = 0 and x^4+ ax + 1 = 0` have a common root, is

Options

A. 2
B. `-2`
C. 3
D. `-3`

Answer

To find the value of \( a \) for which the equations \( x^3 + ax + 1 = 0 \) and \( x^4 + ax + 1 = 0 \) have a common root, we can follow these steps:

### Step 1: Identify the equations
We have two equations:
1. \( x^3 + ax + 1 = 0 \) (Equation 1)
2. \( x^4 + ax + 1 = 0 \) (Equation 2)

### Step 2: Express the second equation in terms of the first
Notice that we can rewrite Equation 2 by factoring out \( x \):
\[ x^4 + ax + 1 = x(x^3) + ax + 1 = 0 \]
This means:
\[ x^4 + ax + 1 = x(x^3 + ax + 1) - (ax + 1) = 0 \]

### Step 3: Set up the subtraction
Subtract Equation 1 from Equation 2:
\[ (x^4 + ax + 1) - (x^3 + ax + 1) = 0 \]
This simplifies to:
\[ x^4 - x^3 = 0 \]

### Step 4: Factor the equation
Factoring out \( x^3 \):
\[ x^3(x - 1) = 0 \]
This gives us two possible roots:
1. \( x^3 = 0 \) which implies \( x = 0 \)
2. \( x - 1 = 0 \) which implies \( x = 1 \)

### Step 5: Substitute the roots back into Equation 1
We will check both roots in Equation 1 to find the corresponding value of \( a \).

#### Case 1: \( x = 0 \)
Substituting \( x = 0 \) into Equation 1:
\[ 0^3 + a(0) + 1 = 0 \implies 1 = 0 \quad \text{(not valid)} \]

#### Case 2: \( x = 1 \)
Substituting \( x = 1 \) into Equation 1:
\[ 1^3 + a(1) + 1 = 0 \implies 1 + a + 1 = 0 \implies a + 2 = 0 \implies a = -2 \]

### Conclusion
The value of \( a \) for which the equations \( x^3 + ax + 1 = 0 \) and \( x^4 + ax + 1 = 0 \) have a common root is:
\[ \boxed{-2} \]