If there is a multiple root of order 3 for the equation `x^4 - 2x ^3 + 2x -a=0` then the other root is

Options

A. `-1`
B. `0`
C. 1
D. 2

Answer

To solve the problem, we need to find the other root of the polynomial equation given that there is a multiple root of order 3. Let's denote the multiple root as \( \alpha \) and the other root as \( \beta \).

The equation is:

\[ x^4 - 2x^3 + 2x - a = 0 \]

### Step 1: Understanding the Roots
Since there is a multiple root of order 3, we can express the roots as \( \alpha, \alpha, \alpha, \beta \).

### Step 2: Using Vieta's Formulas
According to Vieta's formulas, the sum of the roots of the polynomial \( x^4 + bx^3 + cx^2 + dx + e = 0 \) is given by:

\[ \text{Sum of roots} = -\frac{\text{coefficient of } x^3}{\text{coefficient of } x^4} = -(-2) = 2 \]

This means:

\[ 3\alpha + \beta = 2 \tag{1} \]

### Step 3: Sum of the Products of Roots Taken Two at a Time
The sum of the products of the roots taken two at a time is given by:

\[ \text{Sum of products of roots} = \frac{\text{coefficient of } x^2}{\text{coefficient of } x^4} = 0 \]

This means:

\[ \alpha^2 + \alpha^2 + \alpha^2 + 3\alpha\beta = 0 \]

Simplifying gives:

\[ 3\alpha^2 + 3\alpha\beta = 0 \]

Dividing by 3, we have:

\[ \alpha^2 + \alpha\beta = 0 \tag{2} \]

### Step 4: Solving the Equations
From equation (2), we can factor out \( \alpha \):

\[ \alpha(\alpha + \beta) = 0 \]

This gives us two cases:

1. \( \alpha = 0 \)
2. \( \alpha + \beta = 0 \)

#### Case 1: \( \alpha = 0 \)
If \( \alpha = 0 \), substituting into equation (1):

\[ 3(0) + \beta = 2 \implies \beta = 2 \]

#### Case 2: \( \alpha + \beta = 0 \)
If \( \alpha + \beta = 0 \), then \( \beta = -\alpha \). Substituting into equation (1):

\[ 3\alpha - \alpha = 2 \implies 2\alpha = 2 \implies \alpha = 1 \]

Thus, \( \beta = -1 \).

### Conclusion
The possible values for the other root \( \beta \) are:

1. \( \beta = 2 \) when \( \alpha = 0 \)
2. \( \beta = -1 \) when \( \alpha = 1 \)

Therefore, the other root can be either \( 2 \) or \( -1 \).

### Final Answer
The other root is \( \beta = 2 \) or \( \beta = -1 \).

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