Assertion (A ) : The equation whose roots are the squeares of the roots of ` x^4 +x^3+2x^2 +x+1=0` is ` x^4 +3x^3 +4x^2 +3x+1=0`
Reason (R ) : the equation whose roots are the squares of the roots of f (x ) =0 is obtained by eliminating squares root from ` f( sqrt(x))=0`

Options

A. both A and R are true R is the correct explanation of A
B. both A and R are true and R is not correct explanation of A
C. A is true and R is false
D. A is false and R is true

Answer

To solve the problem, we need to verify the assertion and reason provided in the question.

### Step 1: Identify the original polynomial
The original polynomial is given as:
\[ f(x) = x^4 + x^3 + 2x^2 + x + 1 = 0 \]

### Step 2: Substitute \( \sqrt{x} \) into the polynomial
To find the polynomial whose roots are the squares of the roots of \( f(x) \), we substitute \( \sqrt{x} \) into \( f(x) \):
\[ f(\sqrt{x}) = (\sqrt{x})^4 + (\sqrt{x})^3 + 2(\sqrt{x})^2 + \sqrt{x} + 1 \]
This simplifies to:
\[ f(\sqrt{x}) = x^2 + x^{3/2} + 2x + \sqrt{x} + 1 \]

### Step 3: Eliminate the square roots
To eliminate the square roots, we can square both sides of the equation. However, we need to be careful about how we handle the terms. Let's rewrite the equation:
\[ x^2 + x^{3/2} + 2x + \sqrt{x} + 1 = 0 \]

### Step 4: Rearranging and squaring
We can rearrange the equation to isolate the square root terms:
\[ x^2 + 2x + 1 + x^{3/2} + \sqrt{x} = 0 \]
Now, we can square both sides:
\[ (x^2 + 2x + 1)^2 = (x^{3/2} + \sqrt{x})^2 \]

### Step 5: Expand both sides
Expanding the left side:
\[ (x^2 + 2x + 1)^2 = x^4 + 4x^3 + 4x^2 + 4x + 1 \]
Expanding the right side:
\[ (x^{3/2} + \sqrt{x})^2 = x^3 + 2x + 1 \]

### Step 6: Setting the equation
Now we set the two expanded forms equal to each other:
\[ x^4 + 4x^3 + 4x^2 + 4x + 1 = x^3 + 2x + 1 \]

### Step 7: Rearranging the equation
Rearranging gives us:
\[ x^4 + 4x^3 + 4x^2 + 4x + 1 - x^3 - 2x - 1 = 0 \]
This simplifies to:
\[ x^4 + 3x^3 + 4x^2 + 2x + 0 = 0 \]

### Step 8: Final polynomial
Thus, the polynomial whose roots are the squares of the roots of the original polynomial is:
\[ x^4 + 3x^3 + 4x^2 + 3x + 1 = 0 \]

### Conclusion
The assertion is true, and the reasoning provided is correct. Therefore, both the assertion and reason are true, and the reason is the correct explanation for the assertion.

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