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^(4)+1, " "g(x)=x-1`

Answer

To divide the polynomial \( p(x) = x^4 + 1 \) by \( g(x) = x - 1 \) using long division, follow these steps:

### Step 1: Set up the division
Write \( p(x) \) under the long division symbol and \( g(x) \) outside:

\[ \begin{array}{r|l} x - 1 & x^4 + 0x^3 + 0x^2 + 0x + 1 \\ \end{array} \]

### Step 2: Divide the leading terms
Divide the leading term of \( p(x) \) by the leading term of \( g(x) \):

\[ \frac{x^4}{x} = x^3 \]

### Step 3: Multiply and subtract
Multiply \( g(x) \) by \( x^3 \) and subtract from \( p(x) \):

\[ (x - 1) \cdot x^3 = x^4 - x^3 \]

Now subtract:

\[ (x^4 + 0x^3 + 0x^2 + 0x + 1) - (x^4 - x^3) = (0 + x^3 + 0x^2 + 0x + 1) = x^3 + 1 \]

### Step 4: Repeat the process
Now, divide the leading term of the new polynomial \( x^3 + 1 \) by the leading term of \( g(x) \):

\[ \frac{x^3}{x} = x^2 \]

Multiply \( g(x) \) by \( x^2 \) and subtract:

\[ (x - 1) \cdot x^2 = x^3 - x^2 \]

Subtract:

\[ (x^3 + 1) - (x^3 - x^2) = (0 + x^2 + 1) = x^2 + 1 \]

### Step 5: Continue the division
Now, divide the leading term of \( x^2 + 1 \):

\[ \frac{x^2}{x} = x \]

Multiply \( g(x) \) by \( x \) and subtract:

\[ (x - 1) \cdot x = x^2 - x \]

Subtract:

\[ (x^2 + 1) - (x^2 - x) = (0 + x + 1) = x + 1 \]

### Step 6: Final division
Now, divide the leading term of \( x + 1 \):

\[ \frac{x}{x} = 1 \]

Multiply \( g(x) \) by \( 1 \) and subtract:

\[ (x - 1) \cdot 1 = x - 1 \]

Subtract:

\[ (x + 1) - (x - 1) = (0 + 2) = 2 \]

### Conclusion
Now, we cannot divide further since the degree of the remainder \( 2 \) is less than the degree of \( g(x) \).

Thus, the quotient \( q(x) \) and remainder \( r(x) \) are:

\[ q(x) = x^3 + x^2 + x + 1 \]
\[ r(x) = 2 \]