Answer :
To find the remainder when [tex]\(3x^3 - 2x^2 + 4x - 3\)[/tex] is divided by [tex]\(x^2 + 3x + 3\)[/tex], we can use polynomial long division. Here’s how it works step by step:
1. Setup the Division:
We place [tex]\(3x^3 - 2x^2 + 4x - 3\)[/tex] (the dividend) under the division bar and [tex]\(x^2 + 3x + 3\)[/tex] (the divisor) outside.
2. 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], which gives us [tex]\(3x\)[/tex].
3. Multiply and Subtract:
Multiply the entire divisor [tex]\(x^2 + 3x + 3\)[/tex] by [tex]\(3x\)[/tex], which gives [tex]\(3x(x^2 + 3x + 3) = 3x^3 + 9x^2 + 9x\)[/tex].
Subtract this result from the dividend:
[tex]\[
(3x^3 - 2x^2 + 4x - 3) - (3x^3 + 9x^2 + 9x)
\][/tex]
Simplifying, we get:
[tex]\[
-11x^2 - 5x - 3
\][/tex]
4. Repeat the Process:
Now divide [tex]\(-11x^2\)[/tex] by [tex]\(x^2\)[/tex], which gives [tex]\(-11\)[/tex].
5. Multiply and Subtract Again:
Multiply the entire divisor by [tex]\(-11\)[/tex]:
[tex]\[
-11(x^2 + 3x + 3) = -11x^2 - 33x - 33
\][/tex]
Subtract this from [tex]\(-11x^2 - 5x - 3\)[/tex]:
[tex]\[
(-11x^2 - 5x - 3) - (-11x^2 - 33x - 33)
\][/tex]
Simplifying, we get:
[tex]\[
28x + 30
\][/tex]
6. Conclusion:
The degree of the remainder [tex]\(28x + 30\)[/tex] is less than the degree of the divisor [tex]\(x^2 + 3x + 3\)[/tex]. Thus, this is our remainder.
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. Setup the Division:
We place [tex]\(3x^3 - 2x^2 + 4x - 3\)[/tex] (the dividend) under the division bar and [tex]\(x^2 + 3x + 3\)[/tex] (the divisor) outside.
2. 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], which gives us [tex]\(3x\)[/tex].
3. Multiply and Subtract:
Multiply the entire divisor [tex]\(x^2 + 3x + 3\)[/tex] by [tex]\(3x\)[/tex], which gives [tex]\(3x(x^2 + 3x + 3) = 3x^3 + 9x^2 + 9x\)[/tex].
Subtract this result from the dividend:
[tex]\[
(3x^3 - 2x^2 + 4x - 3) - (3x^3 + 9x^2 + 9x)
\][/tex]
Simplifying, we get:
[tex]\[
-11x^2 - 5x - 3
\][/tex]
4. Repeat the Process:
Now divide [tex]\(-11x^2\)[/tex] by [tex]\(x^2\)[/tex], which gives [tex]\(-11\)[/tex].
5. Multiply and Subtract Again:
Multiply the entire divisor by [tex]\(-11\)[/tex]:
[tex]\[
-11(x^2 + 3x + 3) = -11x^2 - 33x - 33
\][/tex]
Subtract this from [tex]\(-11x^2 - 5x - 3\)[/tex]:
[tex]\[
(-11x^2 - 5x - 3) - (-11x^2 - 33x - 33)
\][/tex]
Simplifying, we get:
[tex]\[
28x + 30
\][/tex]
6. Conclusion:
The degree of the remainder [tex]\(28x + 30\)[/tex] is less than the degree of the divisor [tex]\(x^2 + 3x + 3\)[/tex]. Thus, this is our remainder.
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].
