Let `y=(x^(8)+x^(4)+1)/(x^(4)+x^(2)+1)."If"(dy)/(dx)=ax^(3)+bx.` Then,

Options

A. `a=4, b=2`
B. `a=4, b=-2`
C. `a=-2, b=4`
D. none of these

Explanation

To solve the problem, we start with the given function:

\[ y = \frac{x^8 + x^4 + 1}{x^4 + x^2 + 1} \]

We need to find the derivative \( \frac{dy}{dx} \) and express it in the form \( ax^3 + bx \) to determine the values of \( a \) and \( b \).

### Step 1: Differentiate using the Quotient Rule

The Quotient Rule states that if you have a function \( y = \frac{u}{v} \), then the derivative is given by:

\[ \frac{dy}{dx} = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2} \]

Here, let:
- \( u = x^8 + x^4 + 1 \)
- \( v = x^4 + x^2 + 1 \)

Now we need to find \( \frac{du}{dx} \) and \( \frac{dv}{dx} \).

### Step 2: Calculate \( \frac{du}{dx} \) and \( \frac{dv}{dx} \)

1. Differentiate \( u \):
\[ \frac{du}{dx} = 8x^7 + 4x^3 + 0 = 8x^7 + 4x^3 \]

2. Differentiate \( v \):
\[ \frac{dv}{dx} = 4x^3 + 2x + 0 = 4x^3 + 2x \]

### Step 3: Substitute into the Quotient Rule

Now substitute \( u \), \( v \), \( \frac{du}{dx} \), and \( \frac{dv}{dx} \) into the Quotient Rule formula:

\[ \frac{dy}{dx} = \frac{(x^4 + x^2 + 1)(8x^7 + 4x^3) - (x^8 + x^4 + 1)(4x^3 + 2x)}{(x^4 + x^2 + 1)^2} \]

### Step 4: Simplify the expression

To simplify \( \frac{dy}{dx} \), we will expand the numerator:

1. Expand \( (x^4 + x^2 + 1)(8x^7 + 4x^3) \):
- \( x^4 \cdot 8x^7 = 8x^{11} \)
- \( x^2 \cdot 8x^7 = 8x^9 \)
- \( 1 \cdot 8x^7 = 8x^7 \)
- \( x^4 \cdot 4x^3 = 4x^7 \)
- \( x^2 \cdot 4x^3 = 4x^5 \)
- \( 1 \cdot 4x^3 = 4x^3 \)

Combining these gives:
\[ 8x^{11} + 8x^9 + 12x^7 + 4x^5 + 4x^3 \]

2. Expand \( (x^8 + x^4 + 1)(4x^3 + 2x) \):
- \( x^8 \cdot 4x^3 = 4x^{11} \)
- \( x^4 \cdot 4x^3 = 4x^7 \)
- \( 1 \cdot 4x^3 = 4x^3 \)
- \( x^8 \cdot 2x = 2x^9 \)
- \( x^4 \cdot 2x = 2x^5 \)
- \( 1 \cdot 2x = 2x \)

Combining these gives:
\[ 4x^{11} + 2x^9 + 4x^7 + 2x^5 + 2x \]

### Step 5: Combine the results

Now, we can combine the results from both expansions:

\[ \text{Numerator} = (8x^{11} + 8x^9 + 12x^7 + 4x^5 + 4x^3) - (4x^{11} + 2x^9 + 4x^7 + 2x^5 + 2x) \]

This simplifies to:

\[ (8x^{11} - 4x^{11}) + (8x^9 - 2x^9) + (12x^7 - 4x^7) + (4x^5 - 2x^5) + (4x^3) - 2x \]

\[ = 4x^{11} + 6x^9 + 8x^7 + 2x^5 + 4x^3 - 2x \]

### Step 6: Identify coefficients

Now, we need to express \( \frac{dy}{dx} \) in the form \( ax^3 + bx \).

From the simplified numerator, we can see:

- The coefficient of \( x^3 \) is \( 4 \) (from \( 4x^3 \)).
- The coefficient of \( x \) is \( -2 \) (from \( -2x \)).

Thus, we have:

\[ a = 4, \quad b = -2 \]

### Final Answer

The values of \( a \) and \( b \) are:

\[ a = 4, \quad b = -2 \]