To divide the polynomial [tex]\(12x^5 - 36x^4 - 6x^3\)[/tex] by [tex]\(3x^2\)[/tex], you can follow these steps:
1. Divide each term of the numerator by the divisor: [tex]\(3x^2\)[/tex].
Let's break it down term by term:
- The first term is [tex]\(12x^5\)[/tex]:
[tex]\[
\frac{12x^5}{3x^2} = 4x^{5-2} = 4x^3
\][/tex]
- The second term is [tex]\(-36x^4\)[/tex]:
[tex]\[
\frac{-36x^4}{3x^2} = -12x^{4-2} = -12x^2
\][/tex]
- The third term is [tex]\(-6x^3\)[/tex]:
[tex]\[
\frac{-6x^3}{3x^2} = -2x^{3-2} = -2x
\][/tex]
2. Combine the simplified terms:
Therefore, the result of the division is:
[tex]\[
\frac{12x^5 - 36x^4 - 6x^3}{3x^2} = 4x^3 - 12x^2 - 2x
\][/tex]
So, the correct answer is:
[tex]\[ 4x^3 - 12x^2 - 2x \][/tex]
This matches the option:
[tex]\[ \boxed{4x^3 - 12x^2 - 2x} \][/tex]
