Answer :
Let's solve the problem step-by-step to determine the remainder when [tex]\((3x^3 - 2x^2 + 4x - 3)\)[/tex] is divided by [tex]\((x^2 + 3x + 3)\)[/tex].
1. We start by setting up the polynomial division. We need to divide the polynomial [tex]\(3x^3 - 2x^2 + 4x - 3\)[/tex] by [tex]\(x^2 + 3x + 3\)[/tex].
2. Look at the highest degree terms in both the dividend and the divisor. The highest degree term in the dividend is [tex]\(3x^3\)[/tex], and the highest degree term in the divisor is [tex]\(x^2\)[/tex]. To simplify the first step, divide [tex]\(3x^3\)[/tex] by [tex]\(x^2\)[/tex] to get [tex]\(3x\)[/tex].
3. Multiply the entire divisor [tex]\(x^2 + 3x + 3\)[/tex] by [tex]\(3x\)[/tex]:
[tex]\[
3x \cdot (x^2 + 3x + 3) = 3x^3 + 9x^2 + 9x
\][/tex]
4. Subtract this result from the original polynomial:
[tex]\[
(3x^3 - 2x^2 + 4x - 3) - (3x^3 + 9x^2 + 9x)
\][/tex]
Simplify to get the new polynomial:
[tex]\[
(-2x^2 - 9x^2) + (4x - 9x) - 3 = -11x^2 - 5x - 3
\][/tex]
5. Now, divide the new polynomial [tex]\(-11x^2 - 5x - 3\)[/tex] by [tex]\(x^2 + 3x + 3\)[/tex]. The highest degree term is [tex]\(-11x^2\)[/tex], and we need to divide it by [tex]\(x^2\)[/tex]. This gives us [tex]\(-11\)[/tex].
6. Multiply the entire divisor [tex]\(x^2 + 3x + 3\)[/tex] by [tex]\(-11\)[/tex]:
[tex]\[
-11 \cdot (x^2 + 3x + 3) = -11x^2 - 33x - 33
\][/tex]
7. Subtract this result from our new polynomial:
[tex]\[
(-11x^2 - 5x - 3) - (-11x^2 - 33x - 33)
\][/tex]
Simplify to get the remainder:
[tex]\[
(-5x + 33x) + (-3 + 33) = 28x + 30
\][/tex]
So, the remainder when [tex]\(3x^3 - 2x^2 + 4x - 3\)[/tex] is divided by [tex]\(x^2 + 3x + 3\)[/tex] is [tex]\(\boxed{28x + 30}\)[/tex].
1. We start by setting up the polynomial division. We need to divide the polynomial [tex]\(3x^3 - 2x^2 + 4x - 3\)[/tex] by [tex]\(x^2 + 3x + 3\)[/tex].
2. Look at the highest degree terms in both the dividend and the divisor. The highest degree term in the dividend is [tex]\(3x^3\)[/tex], and the highest degree term in the divisor is [tex]\(x^2\)[/tex]. To simplify the first step, divide [tex]\(3x^3\)[/tex] by [tex]\(x^2\)[/tex] to get [tex]\(3x\)[/tex].
3. Multiply the entire divisor [tex]\(x^2 + 3x + 3\)[/tex] by [tex]\(3x\)[/tex]:
[tex]\[
3x \cdot (x^2 + 3x + 3) = 3x^3 + 9x^2 + 9x
\][/tex]
4. Subtract this result from the original polynomial:
[tex]\[
(3x^3 - 2x^2 + 4x - 3) - (3x^3 + 9x^2 + 9x)
\][/tex]
Simplify to get the new polynomial:
[tex]\[
(-2x^2 - 9x^2) + (4x - 9x) - 3 = -11x^2 - 5x - 3
\][/tex]
5. Now, divide the new polynomial [tex]\(-11x^2 - 5x - 3\)[/tex] by [tex]\(x^2 + 3x + 3\)[/tex]. The highest degree term is [tex]\(-11x^2\)[/tex], and we need to divide it by [tex]\(x^2\)[/tex]. This gives us [tex]\(-11\)[/tex].
6. Multiply the entire divisor [tex]\(x^2 + 3x + 3\)[/tex] by [tex]\(-11\)[/tex]:
[tex]\[
-11 \cdot (x^2 + 3x + 3) = -11x^2 - 33x - 33
\][/tex]
7. Subtract this result from our new polynomial:
[tex]\[
(-11x^2 - 5x - 3) - (-11x^2 - 33x - 33)
\][/tex]
Simplify to get the remainder:
[tex]\[
(-5x + 33x) + (-3 + 33) = 28x + 30
\][/tex]
So, the remainder when [tex]\(3x^3 - 2x^2 + 4x - 3\)[/tex] is divided by [tex]\(x^2 + 3x + 3\)[/tex] is [tex]\(\boxed{28x + 30}\)[/tex].
