If `y=(x^4+x^2+1)/(x^2+x+1)` then `(dy)/(dx)=a x+b` , find `a` and `b`

Options

A. `a=2,b=1`
B. `a=-2,b=1`
C. `a=2,b=-1`
D. `a=-2,b=-1`

Answer

To find the values of \( a \) and \( b \) in the expression \( \frac{dy}{dx} = ax + b \) for the function \( y = \frac{x^4 + x^2 + 1}{x^2 + x + 1} \), we will follow these steps:

### Step 1: Rewrite the function
We start with the function:
\[ y = \frac{x^4 + x^2 + 1}{x^2 + x + 1} \]

### Step 2: Simplify the numerator
We can manipulate the numerator by adding and subtracting \( x^2 \):
\[ y = \frac{x^4 + 2x^2 + 1 - x^2}{x^2 + x + 1} \]
This gives us:
\[ y = \frac{(x^2 + 1)^2 - x^2}{x^2 + x + 1} \]

### Step 3: Factor the numerator
Using the difference of squares, we can factor the numerator:
\[ y = \frac{(x^2 + 1 - x)(x^2 + 1 + x)}{x^2 + x + 1} \]

### Step 4: Cancel common terms
Notice that \( x^2 + x + 1 \) is present in both the numerator and denominator:
\[ y = x^2 + 1 - x \]
Thus, we simplify to:
\[ y = x^2 - x + 1 \]

### Step 5: Differentiate the function
Now we differentiate \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = \frac{d}{dx}(x^2 - x + 1) = 2x - 1 \]

### Step 6: Compare with the given form
We have:
\[ \frac{dy}{dx} = 2x - 1 \]
This can be compared to the form \( ax + b \):
- Here, \( a = 2 \)
- And \( b = -1 \)

### Final Answer
Thus, the values of \( a \) and \( b \) are:
\[ a = 2, \quad b = -1 \]