If `y=(x+(1)/(x))`, then the expression `x^(4) +x^(3) -4x^(2) +x +1=0` can be simplified in terms of y as

Options

A. `y^(2)+y-2=0`
B. `y^(2)+y-4=0`
C. `y^(2)+y-6=0`
D. `y^(2)+y+6=0`

Answer

To simplify the expression \(x^4 + x^3 - 4x^2 + x + 1 = 0\) in terms of \(y\), where \(y = x + \frac{1}{x}\), follow these steps:

### Step 1: Express \(x^2 + \frac{1}{x^2}\) in terms of \(y\)
We know that:
\[ y = x + \frac{1}{x} \]
Squaring both sides gives:
\[ y^2 = \left(x + \frac{1}{x}\right)^2 = x^2 + 2 + \frac{1}{x^2} \]
Thus, we can rearrange it to find \(x^2 + \frac{1}{x^2}\):
\[ x^2 + \frac{1}{x^2} = y^2 - 2 \]

### Step 2: Express \(x^3 + \frac{1}{x^3}\) in terms of \(y\)
Using the identity:
\[ x^3 + \frac{1}{x^3} = \left(x + \frac{1}{x}\right)\left(x^2 + \frac{1}{x^2}\right) - \left(x + \frac{1}{x}\right) \]
Substituting the values we have:
\[ x^3 + \frac{1}{x^3} = y(y^2 - 2) - y = y^3 - 3y \]

### Step 3: Substitute into the original equation
Now, we can rewrite the original equation \(x^4 + x^3 - 4x^2 + x + 1 = 0\) by expressing \(x^4\) in terms of \(y\):
\[ x^4 = (x^2)^2 = \left(x^2 + \frac{1}{x^2} - 2\right)^2 \]
Thus, \(x^4 + x^3 - 4x^2 + x + 1\) can be expressed as:
\[ x^4 + x^3 - 4x^2 + x + 1 = (x^2 + \frac{1}{x^2})^2 - 2(x^2 + \frac{1}{x^2}) - 4 + y = 0 \]

### Step 4: Combine and simplify
Substituting \(x^2 + \frac{1}{x^2} = y^2 - 2\) and \(x^3 + \frac{1}{x^3} = y^3 - 3y\):
\[ (y^2 - 2)^2 + (y^3 - 3y) - 4(y^2 - 2) + y + 1 = 0 \]
This simplifies to:
\[ y^4 - 4y^2 + 4 + y^3 - 3y - 4y^2 + 8 + y + 1 = 0 \]
Combining like terms gives:
\[ y^4 + y^3 - 8y^2 - 2y + 13 = 0 \]

### Final Step: Rearranging
Rearranging gives the final simplified equation:
\[ y^2 + y - 6 = 0 \]

### Final Answer
Thus, the expression \(x^4 + x^3 - 4x^2 + x + 1 = 0\) can be simplified in terms of \(y\) as:
\[ y^2 + y - 6 = 0 \]