Answer :
We want to divide the polynomial
[tex]$$
3x^3 - 2x^2 + 4x - 3
$$[/tex]
by
[tex]$$
x^2 + 3x + 3.
$$[/tex]
Step 1. Divide the Leading Terms
Divide the leading term of the dividend, [tex]$3x^3$[/tex], by the leading term of the divisor, [tex]$x^2$[/tex]:
[tex]$$
\frac{3x^3}{x^2} = 3x.
$$[/tex]
So, the first term of the quotient is [tex]$3x$[/tex].
Step 2. Multiply and Subtract
Multiply the divisor by [tex]$3x$[/tex]:
[tex]$$
3x \cdot (x^2 + 3x + 3) = 3x^3 + 9x^2 + 9x.
$$[/tex]
Subtract this from the original polynomial:
[tex]\[
\begin{aligned}
(3x^3 - 2x^2 + 4x - 3) - (3x^3 + 9x^2 + 9x) &= (3x^3 - 3x^3) + (-2x^2 - 9x^2) \\
&\quad + (4x - 9x) - 3 \\
&= -11x^2 - 5x - 3.
\end{aligned}
\][/tex]
Step 3. Repeat the Process
Now, consider the new polynomial [tex]$-11x^2 - 5x - 3$[/tex]. Divide its leading term [tex]$-11x^2$[/tex] by the leading term of the divisor [tex]$x^2$[/tex]:
[tex]$$
\frac{-11x^2}{x^2} = -11.
$$[/tex]
So, the next term in the quotient is [tex]$-11$[/tex], making the complete quotient:
[tex]$$
3x - 11.
$$[/tex]
Step 4. Multiply and Subtract Again
Multiply the divisor by [tex]$-11$[/tex]:
[tex]$$
-11 \cdot (x^2 + 3x + 3) = -11x^2 - 33x - 33.
$$[/tex]
Subtract this from the current polynomial:
[tex]\[
\begin{aligned}
(-11x^2 - 5x - 3) - (-11x^2 - 33x - 33) &= (-11x^2 + 11x^2) + (-5x + 33x) \\
&\quad + (-3 + 33) \\
&= 28x + 30.
\end{aligned}
\][/tex]
Step 5. Conclude the Division
Since the degree of [tex]$28x + 30$[/tex] is less than the degree of the divisor [tex]$x^2 + 3x + 3$[/tex], the division stops here. Thus, the quotient is
[tex]$$
3x - 11
$$[/tex]
and the remainder is
[tex]$$
28x + 30.
$$[/tex]
Final Answer:
The remainder when [tex]$\left(3x^3 - 2x^2 + 4x - 3\right)$[/tex] is divided by [tex]$\left(x^2 + 3x + 3\right)$[/tex] is
[tex]$$
\boxed{28x + 30}.
$$[/tex]
[tex]$$
3x^3 - 2x^2 + 4x - 3
$$[/tex]
by
[tex]$$
x^2 + 3x + 3.
$$[/tex]
Step 1. Divide the Leading Terms
Divide the leading term of the dividend, [tex]$3x^3$[/tex], by the leading term of the divisor, [tex]$x^2$[/tex]:
[tex]$$
\frac{3x^3}{x^2} = 3x.
$$[/tex]
So, the first term of the quotient is [tex]$3x$[/tex].
Step 2. Multiply and Subtract
Multiply the divisor by [tex]$3x$[/tex]:
[tex]$$
3x \cdot (x^2 + 3x + 3) = 3x^3 + 9x^2 + 9x.
$$[/tex]
Subtract this from the original polynomial:
[tex]\[
\begin{aligned}
(3x^3 - 2x^2 + 4x - 3) - (3x^3 + 9x^2 + 9x) &= (3x^3 - 3x^3) + (-2x^2 - 9x^2) \\
&\quad + (4x - 9x) - 3 \\
&= -11x^2 - 5x - 3.
\end{aligned}
\][/tex]
Step 3. Repeat the Process
Now, consider the new polynomial [tex]$-11x^2 - 5x - 3$[/tex]. Divide its leading term [tex]$-11x^2$[/tex] by the leading term of the divisor [tex]$x^2$[/tex]:
[tex]$$
\frac{-11x^2}{x^2} = -11.
$$[/tex]
So, the next term in the quotient is [tex]$-11$[/tex], making the complete quotient:
[tex]$$
3x - 11.
$$[/tex]
Step 4. Multiply and Subtract Again
Multiply the divisor by [tex]$-11$[/tex]:
[tex]$$
-11 \cdot (x^2 + 3x + 3) = -11x^2 - 33x - 33.
$$[/tex]
Subtract this from the current polynomial:
[tex]\[
\begin{aligned}
(-11x^2 - 5x - 3) - (-11x^2 - 33x - 33) &= (-11x^2 + 11x^2) + (-5x + 33x) \\
&\quad + (-3 + 33) \\
&= 28x + 30.
\end{aligned}
\][/tex]
Step 5. Conclude the Division
Since the degree of [tex]$28x + 30$[/tex] is less than the degree of the divisor [tex]$x^2 + 3x + 3$[/tex], the division stops here. Thus, the quotient is
[tex]$$
3x - 11
$$[/tex]
and the remainder is
[tex]$$
28x + 30.
$$[/tex]
Final Answer:
The remainder when [tex]$\left(3x^3 - 2x^2 + 4x - 3\right)$[/tex] is divided by [tex]$\left(x^2 + 3x + 3\right)$[/tex] is
[tex]$$
\boxed{28x + 30}.
$$[/tex]
