College

What is the remainder when [tex]3x^3 - 2x^2 + 4x - 3[/tex] is divided by [tex]x^2 + 3x + 3[/tex]?

A. 30
B. [tex]3x - 11[/tex]
C. [tex]28x - 36[/tex]
D. [tex]28x + 30[/tex]

Answer :

To find the remainder when the polynomial [tex]\(3x^3 - 2x^2 + 4x - 3\)[/tex] is divided by [tex]\(x^2 + 3x + 3\)[/tex], we use polynomial long division.

### Step-by-Step Solution:

1. Divide the first term of the dividend by the first term of the divisor:

- The first term of the dividend is [tex]\(3x^3\)[/tex] and the first term of the divisor is [tex]\(x^2\)[/tex].
- Divide [tex]\(3x^3\)[/tex] by [tex]\(x^2\)[/tex] to get [tex]\(3x\)[/tex].

2. Multiply the entire divisor by this term:

- Multiply [tex]\(3x\)[/tex] by [tex]\(x^2 + 3x + 3\)[/tex].
- This gives you [tex]\(3x \cdot (x^2 + 3x + 3) = 3x^3 + 9x^2 + 9x\)[/tex].

3. Subtract this from the original polynomial:

- Subtract [tex]\(3x^3 + 9x^2 + 9x\)[/tex] from [tex]\(3x^3 - 2x^2 + 4x - 3\)[/tex].
- The result is: [tex]\((-2x^2 + 4x - 3) - (9x^2 + 9x) = -11x^2 - 5x - 3\)[/tex].

4. Repeat the process with the new polynomial:

- Divide the first term of the new polynomial, [tex]\(-11x^2\)[/tex], by the first term of the divisor, [tex]\(x^2\)[/tex].
- This gives you [tex]\(-11\)[/tex].

5. Multiply the entire divisor by [tex]\(-11\)[/tex]:

- Multiply [tex]\(-11\)[/tex] by [tex]\(x^2 + 3x + 3\)[/tex].
- This gives you [tex]\(-11 \cdot (x^2 + 3x + 3) = -11x^2 - 33x - 33\)[/tex].

6. Subtract this from the current polynomial:

- Subtract [tex]\(-11x^2 - 33x - 33\)[/tex] from [tex]\(-11x^2 - 5x - 3\)[/tex].
- The result is: [tex]\((-11x^2 - 5x - 3) - (-11x^2 - 33x - 33)\)[/tex].
- Simplify to get [tex]\(28x + 30\)[/tex].

The remainder when [tex]\(3x^3 - 2x^2 + 4x - 3\)[/tex] is divided by [tex]\(x^2 + 3x + 3\)[/tex] is:

[tex]\[28x + 30\][/tex]

So the answer is [tex]\(28x + 30\)[/tex].