Divide p(x) by g(x) in each of the following questions and find the quotient q(x) and remainder r(x) :
`p(x)=3x^(3)-4x^(2)+2x+5, " " g(x)=x+3`

Answer

To divide the polynomial \( p(x) = 3x^3 - 4x^2 + 2x + 5 \) by \( g(x) = x + 3 \), we will use polynomial long division. Here are the steps to find the quotient \( q(x) \) and the remainder \( r(x) \):

### Step 1: Set up the division
We write \( p(x) \) as the dividend and \( g(x) \) as the divisor:

\[ \text{Dividend: } p(x) = 3x^3 - 4x^2 + 2x + 5 \]
\[ \text{Divisor: } g(x) = x + 3 \]

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

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

### Step 3: Multiply and subtract
Multiply \( g(x) \) by \( 3x^2 \):

\[ 3x^2 \cdot (x + 3) = 3x^3 + 9x^2 \]

Now, subtract this from \( p(x) \):

\[ (3x^3 - 4x^2 + 2x + 5) - (3x^3 + 9x^2) = -4x^2 - 9x^2 + 2x + 5 = -13x^2 + 2x + 5 \]

### Step 4: Repeat the process
Now, take the new polynomial \( -13x^2 + 2x + 5 \) and divide the leading term by the leading term of \( g(x) \):

\[ \frac{-13x^2}{x} = -13x \]

Multiply \( g(x) \) by \( -13x \):

\[ -13x \cdot (x + 3) = -13x^2 - 39x \]

Subtract this from the previous result:

\[ (-13x^2 + 2x + 5) - (-13x^2 - 39x) = 2x + 39x + 5 = 41x + 5 \]

### Step 5: Repeat again
Now, divide the leading term \( 41x \) by the leading term of \( g(x) \):

\[ \frac{41x}{x} = 41 \]

Multiply \( g(x) \) by \( 41 \):

\[ 41 \cdot (x + 3) = 41x + 123 \]

Subtract this from the previous result:

\[ (41x + 5) - (41x + 123) = 5 - 123 = -118 \]

### Step 6: Conclusion
Now, we have completed the division. The quotient \( q(x) \) and remainder \( r(x) \) are:

\[ q(x) = 3x^2 - 13x + 41 \]
\[ r(x) = -118 \]

### Final Answer
Thus, the quotient is \( q(x) = 3x^2 - 13x + 41 \) and the remainder is \( r(x) = -118 \).

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