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^(3)+3x^(2)+2x+1, " " g(x)=x+2`

Answer

To divide the polynomial \( p(x) = x^3 + 3x^2 + 2x + 1 \) by \( g(x) = x + 2 \), we will use polynomial long division. Here are the steps:

### Step 1: Set up the division
We write \( p(x) \) as the dividend and \( g(x) \) as the divisor:
\[ \text{Dividend: } p(x) = x^3 + 3x^2 + 2x + 1 \]
\[ \text{Divisor: } g(x) = x + 2 \]

### Step 2: Divide the leading terms
Divide the leading term of the dividend by the leading term of the divisor:
\[ \frac{x^3}{x} = x^2 \]
This gives us the first term of the quotient \( q(x) \).

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

### Step 4: Repeat the process
Now we have a new polynomial \( x^2 + 2x + 1 \). We will repeat the process:
Divide the leading term:
\[ \frac{x^2}{x} = x \]
This gives us the next term of the quotient \( q(x) \).

### Step 5: Multiply and subtract again
Multiply \( g(x) \) by \( x \):
\[ g(x) \cdot x = (x + 2) \cdot x = x^2 + 2x \]
Now, subtract this from the new polynomial:
\[ (x^2 + 2x + 1) - (x^2 + 2x) = 1 \]

### Step 6: Finalize the quotient and remainder
Now we have a remainder of \( 1 \). Since \( 1 \) cannot be divided by \( x + 2 \), we stop here.

### Conclusion
The quotient \( q(x) \) and remainder \( r(x) \) are:
\[ q(x) = x^2 + x \]
\[ r(x) = 1 \]