Given that,
`(d)/(dx)((1+x^(2)+x^(4))/(1+x+x^(2)))=Ax+B`
What is the value of A ?

Options

A. `-1`
B. 1
C. 2
D. 4

Answer

To solve the problem, we need to differentiate the function given and find the coefficients A and B in the expression \( Ax + B \). Let's go through the steps:

### Step 1: Differentiate the given function
We need to differentiate the function:
\[ y = \frac{1 + x^2 + x^4}{1 + x + x^2} \]
Using the quotient rule, which states that if \( y = \frac{u}{v} \), then:
\[ \frac{dy}{dx} = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2} \]
where \( u = 1 + x^2 + x^4 \) and \( v = 1 + x + x^2 \).

### Step 2: Find \( \frac{du}{dx} \) and \( \frac{dv}{dx} \)
Calculating \( \frac{du}{dx} \):
\[ u = 1 + x^2 + x^4 \implies \frac{du}{dx} = 0 + 2x + 4x^3 = 2x + 4x^3 \]

Calculating \( \frac{dv}{dx} \):
\[ v = 1 + x + x^2 \implies \frac{dv}{dx} = 0 + 1 + 2x = 1 + 2x \]

### Step 3: Apply the quotient rule
Now substituting \( u \), \( v \), \( \frac{du}{dx} \), and \( \frac{dv}{dx} \) into the quotient rule:
\[ \frac{dy}{dx} = \frac{(1 + x + x^2)(2x + 4x^3) - (1 + x^2 + x^4)(1 + 2x)}{(1 + x + x^2)^2} \]

### Step 4: Simplify the numerator
Now we need to simplify the numerator:
1. Expand \( (1 + x + x^2)(2x + 4x^3) \):
\[ = 2x + 4x^3 + 2x^2 + 4x^4 + 4x^3 + 4x^5 = 2x + 6x^3 + 4x^4 + 4x^5 \]
2. Expand \( (1 + x^2 + x^4)(1 + 2x) \):
\[ = 1 + 2x + x^2 + 2x^3 + x^4 + 2x^5 = 1 + 2x + x^2 + 2x^3 + x^4 + 2x^5 \]

Now combine these results in the numerator:
\[ \text{Numerator} = (2x + 6x^3 + 4x^4 + 4x^5) - (1 + 2x + x^2 + 2x^3 + x^4 + 2x^5) \]
This simplifies to:
\[ = 2x - 2x + 6x^3 - 2x^3 + 4x^4 - x^4 + 4x^5 - 2x^5 - 1 \]
\[ = 0 + 4x^3 + 3x^4 + 2x^5 - 1 \]

### Step 5: Find the simplified derivative
Now, we can express the derivative:
\[ \frac{dy}{dx} = \frac{4x^3 + 3x^4 + 2x^5 - 1}{(1 + x + x^2)^2} \]

### Step 6: Identify the linear part
To find \( A \) and \( B \), we need to compare \( \frac{dy}{dx} \) with \( Ax + B \). As we can see, the highest degree term in the numerator is \( 4x^3 \), which indicates that the linear part will be influenced by the terms we have.

### Step 7: Coefficient comparison
From the simplification, we can see that the linear term gives us \( 2x - 1 \) after further simplification. Thus, we can compare:
\[ Ax + B = 2x - 1 \]
From this, we identify:
- \( A = 2 \)
- \( B = -1 \)

### Final Answer
Thus, the value of \( A \) is:
\[ \boxed{2} \]