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)+4x^(2)+2, " "g(x)=x^(2)+1`

Answer

To divide the polynomial \( p(x) = x^4 + 4x^2 + 2 \) by \( g(x) = x^2 + 1 \), we will perform polynomial long division. Here are the steps:

### Step 1: Set up the division
We will divide \( p(x) \) by \( g(x) \):
\[ \text{Divide } p(x) = x^4 + 4x^2 + 2 \text{ by } g(x) = x^2 + 1. \]

### Step 2: Divide the leading terms
Divide the leading term of \( p(x) \) by the leading term of \( g(x) \):
\[ \frac{x^4}{x^2} = x^2. \]
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 \( p(x) \):
\[ g(x) \cdot x^2 = (x^2 + 1) \cdot x^2 = x^4 + x^2. \]
Now subtract this from \( p(x) \):
\[ (x^4 + 4x^2 + 2) - (x^4 + x^2) = (4x^2 - x^2) + 2 = 3x^2 + 2. \]

### Step 4: Repeat the process
Now, we have a new polynomial \( 3x^2 + 2 \). Divide the leading term of this polynomial by the leading term of \( g(x) \):
\[ \frac{3x^2}{x^2} = 3. \]
This gives us the next term of the quotient \( q(x) \).

### Step 5: Multiply and subtract again
Now, multiply \( g(x) \) by \( 3 \) and subtract:
\[ g(x) \cdot 3 = (x^2 + 1) \cdot 3 = 3x^2 + 3. \]
Now subtract this from \( 3x^2 + 2 \):
\[ (3x^2 + 2) - (3x^2 + 3) = 2 - 3 = -1. \]

### Step 6: Write the final result
Now we have completed the division. The quotient \( q(x) \) and remainder \( r(x) \) are:
\[ q(x) = x^2 + 3, \]
\[ r(x) = -1. \]

### Final Answer
The quotient is \( q(x) = x^2 + 3 \) and the remainder is \( r(x) = -1 \).

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