If `(d)/(dx)((1+x^2+x^4)/(1+x+x^2)) = a+bx` , find the values of a and b .

Answer

To solve the problem, we need to differentiate the function \( \frac{1 + x^2 + x^4}{1 + x + x^2} \) and express the result in the form \( a + bx \).

### Step 1: Differentiate the function
We will use the quotient rule for differentiation, which states that if we have a function \( \frac{u}{v} \), then its derivative is given by:

\[ \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2} \]

Here, let:
- \( u = 1 + x^2 + x^4 \)
- \( v = 1 + x + x^2 \)

### Step 2: Calculate \( \frac{du}{dx} \) and \( \frac{dv}{dx} \)
Now we need to find the derivatives of \( u \) and \( v \):
- \( \frac{du}{dx} = 0 + 2x + 4x^3 = 2x + 4x^3 \)
- \( \frac{dv}{dx} = 0 + 1 + 2x = 1 + 2x \)

### Step 3: Apply the quotient rule
Now we can apply the quotient rule:

\[ \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{(1 + x + x^2)(2x + 4x^3) - (1 + x^2 + x^4)(1 + 2x)}{(1 + x + x^2)^2} \]

### Step 4: Simplify the numerator
Now we will simplify the numerator step by step:

1. Expand \( (1 + x + x^2)(2x + 4x^3) \):
\[ = 2x + 4x^3 + 2x^2 + 4x^4 \]
Combine like terms:
\[ = 4x^4 + 4x^3 + 2x^2 + 2x \]

2. Expand \( (1 + x^2 + x^4)(1 + 2x) \):
\[ = 1 + 2x + x^2 + 2x^3 + x^4 + 2x^5 \]

### Step 5: Combine and simplify
Now we will subtract the second expansion from the first:
\[ (4x^4 + 4x^3 + 2x^2 + 2x) - (1 + 2x + x^2 + 2x^3 + x^4 + 2x^5) \]

Combining like terms:
- For \( x^5 \): \( -2x^5 \)
- For \( x^4 \): \( 4x^4 - x^4 = 3x^4 \)
- For \( x^3 \): \( 4x^3 - 2x^3 = 2x^3 \)
- For \( x^2 \): \( 2x^2 - x^2 = x^2 \)
- For \( x \): \( 2x - 2x = 0 \)
- Constant term: \( -1 \)

Thus, the numerator simplifies to:
\[ -2x^5 + 3x^4 + 2x^3 + x^2 - 1 \]

### Step 6: Find the derivative in the form \( a + bx \)
Since we are interested in the expression \( a + bx \), we need to focus on the terms that will remain after simplification. As \( x \to 0 \), the higher-order terms will vanish, and we can evaluate the derivative at \( x = 0 \).

### Step 7: Evaluate at \( x = 0 \)
To find \( a \) and \( b \), we can evaluate the simplified expression at \( x = 0 \):
\[ \frac{d}{dx}\left(\frac{1 + x^2 + x^4}{1 + x + x^2}\right) \text{ at } x = 0 \]

The derivative simplifies to:
\[ \frac{0 - 1}{1^2} = -1 \]
This means \( a = -1 \) and \( b = 0 \).

### Final Answer
Thus, the values of \( a \) and \( b \) are:
- \( a = -1 \)
- \( b = 0 \)