Divide :
`16 + 8x + x^(6) - 8x^(3) - 2x^(4) + x^(2)` by `x + 4 - x^(3)`

Answer

To divide the polynomial \( 16 + 8x + x^6 - 8x^3 - 2x^4 + x^2 \) by \( x + 4 - x^3 \), we will use the long division method. Here’s the step-by-step solution:

### Step 1: Arrange the Polynomials
First, we need to write both the dividend and the divisor in standard form (descending order of powers).

**Dividend:**
\[ x^6 - 2x^4 - 8x^3 + x^2 + 8x + 16 \]

**Divisor:**
\[ -x^3 + x + 4 \]

### Step 2: Divide the Leading Terms
Next, we divide the leading term of the dividend by the leading term of the divisor.

\[ \frac{x^6}{-x^3} = -x^3 \]

### Step 3: Multiply and Subtract
Now, we multiply the entire divisor by \(-x^3\) and subtract the result from the dividend.

**Multiplication:**
\[ -x^3(-x^3 + x + 4) = x^6 - x^4 - 4x^3 \]

**Subtraction:**
\[ (x^6 - 2x^4 - 8x^3 + x^2 + 8x + 16) - (x^6 - x^4 - 4x^3) \]
This simplifies to:
\[ (-2x^4 + x^4) + (-8x^3 + 4x^3) + x^2 + 8x + 16 = -x^4 - 4x^3 + x^2 + 8x + 16 \]

### Step 4: Repeat the Process
Now we repeat the process with the new polynomial \(-x^4 - 4x^3 + x^2 + 8x + 16\).

**Divide the leading terms:**
\[ \frac{-x^4}{-x^3} = x \]

**Multiply and Subtract:**
\[ x(-x^3 + x + 4) = -x^4 + x^2 + 4x \]

Now we subtract:
\[ (-x^4 - 4x^3 + x^2 + 8x + 16) - (-x^4 + x^2 + 4x) \]
This simplifies to:
\[ (-4x^3 + 8x - 4x + 16) = -4x^3 + 4x + 16 \]

### Step 5: Continue the Division
Now we continue with \(-4x^3 + 4x + 16\).

**Divide the leading terms:**
\[ \frac{-4x^3}{-x^3} = 4 \]

**Multiply and Subtract:**
\[ 4(-x^3 + x + 4) = -4x^3 + 4x + 16 \]

Now we subtract:
\[ (-4x^3 + 4x + 16) - (-4x^3 + 4x + 16) = 0 \]

### Final Result
Since the remainder is 0, we conclude that:
\[ \text{The quotient is } -x^3 + x + 4 \]