Divide p(x) by g(x) in each of the following questions and find the quotient q(x) and remainder r(x) :
`p(x)=x^(6)-1, " " g(x)=x^(2)+1`

Answer

To divide the polynomial \( p(x) = x^6 - 1 \) by \( g(x) = x^2 + 1 \) and find the quotient \( q(x) \) and the remainder \( r(x) \), we will use the long division method for polynomials. Here’s a step-by-step solution:

### Step 1: Set up the division
We write \( p(x) \) as the dividend and \( g(x) \) as the divisor:
\[ \text{Dividend: } p(x) = x^6 - 1 \]
\[ \text{Divisor: } g(x) = x^2 + 1 \]

### Step 2: Divide the leading term
Divide the leading term of the dividend by the leading term of the divisor:
\[ \frac{x^6}{x^2} = x^4 \]
This gives us the first term of the quotient \( q(x) \).

### Step 3: Multiply and subtract
Now, multiply \( g(x) \) by \( x^4 \) and subtract from \( p(x) \):
\[ (x^2 + 1) \cdot x^4 = x^6 + x^4 \]
Now subtract this from \( p(x) \):
\[ (x^6 - 1) - (x^6 + x^4) = -x^4 - 1 \]

### Step 4: Repeat the process
Now, we have a new polynomial \( -x^4 - 1 \). Divide the leading term by the leading term of the divisor:
\[ \frac{-x^4}{x^2} = -x^2 \]
Add this to the quotient \( q(x) \):
\[ q(x) = x^4 - x^2 \]

### Step 5: Multiply and subtract again
Multiply \( g(x) \) by \( -x^2 \):
\[ (x^2 + 1) \cdot (-x^2) = -x^4 - x^2 \]
Subtract this from \( -x^4 - 1 \):
\[ (-x^4 - 1) - (-x^4 - x^2) = x^2 - 1 \]

### Step 6: Continue the process
Now we have \( x^2 - 1 \). Divide the leading term:
\[ \frac{x^2}{x^2} = 1 \]
Add this to the quotient:
\[ q(x) = x^4 - x^2 + 1 \]

### Step 7: Multiply and subtract one last time
Multiply \( g(x) \) by \( 1 \):
\[ (x^2 + 1) \cdot 1 = x^2 + 1 \]
Subtract this from \( x^2 - 1 \):
\[ (x^2 - 1) - (x^2 + 1) = -2 \]

### Final Result
Now we have:
- Quotient:
\[ q(x) = x^4 - x^2 + 1 \]
- Remainder:
\[ r(x) = -2 \]

### Summary
Thus, when dividing \( p(x) \) by \( g(x) \), we find:
\[ \text{Quotient } q(x) = x^4 - x^2 + 1 \]
\[ \text{Remainder } r(x) = -2 \]