To find the remainder when dividing the polynomial [tex]\(6x^3 + 2x^2 + 2\)[/tex] by [tex]\(x + 2\)[/tex], we can use polynomial long division or synthetic division. Here is a step-by-step breakdown:
1. Setup the division problem:
- Dividend (the polynomial to be divided): [tex]\(6x^3 + 2x^2 + 2\)[/tex]
- Divisor: [tex]\(x + 2\)[/tex]
2. Divide the leading term of the dividend by the leading term of the divisor:
- The leading term of the dividend is [tex]\(6x^3\)[/tex] and the leading term of the divisor is [tex]\(x\)[/tex].
- [tex]\( \frac{6x^3}{x} = 6x^2 \)[/tex]
3. Multiply the entire divisor [tex]\(x + 2\)[/tex] by the result from step 2:
- [tex]\( 6x^2 \times (x + 2) = 6x^3 + 12x^2 \)[/tex]
4. Subtract the result from step 3 from the original polynomial:
- [tex]\( (6x^3 + 2x^2 + 2) - (6x^3 + 12x^2) \)[/tex]
- [tex]\( 6x^3 + 2x^2 + 2 - 6x^3 - 12x^2 = -10x^2 + 2 \)[/tex]
5. Repeat the process with the new polynomial [tex]\(-10x^2 + 2\)[/tex]:
- Divide the leading term [tex]\(-10x^2\)[/tex] by the leading term of the divisor [tex]\(x\)[/tex].
- [tex]\( \frac{-10x^2}{x} = -10x \)[/tex]
6. Multiply the entire divisor by the result from the previous step:
- [tex]\( -10x \times (x + 2) = -10x^2 - 20x \)[/tex]
7. Subtract this result from the current polynomial:
- [tex]\( (-10x^2 + 2) - (-10x^2 - 20x) \)[/tex]
- [tex]\( -10x^2 + 2 + 10x^2 + 20x = 20x + 2 \)[/tex]
8. Repeat the process again with the new polynomial [tex]\(20x + 2\)[/tex]:
- Divide the leading term [tex]\(20x\)[/tex] by the leading term of the divisor [tex]\(x\)[/tex].
- [tex]\( \frac{20x}{x} = 20 \)[/tex]
9. Multiply the entire divisor by the result from the previous step:
- [tex]\( 20 \times (x + 2) = 20x + 40 \)[/tex]
10. Subtract this result from the current polynomial:
- [tex]\( (20x + 2) - (20x + 40) \)[/tex]
- [tex]\( 20x + 2 - 20x - 40 = 2 - 40 = -38 \)[/tex]
Thus, the remainder when dividing the polynomial [tex]\(6x^3 + 2x^2 + 2\)[/tex] by [tex]\(x + 2\)[/tex] is [tex]\(-38\)[/tex].
So, the correct answer is:
D. [tex]\(-38\)[/tex]
