Divide [tex]\(12x^5 - 36x^4 - 6x^3\)[/tex] by [tex]\(3x^2\)[/tex].

A. [tex]\(4x^2 + 12x + 2\)[/tex]

B. [tex]\(4x^2 - 12x - 2\)[/tex]

C. [tex]\(4x^3 + 12x^2 + 2x\)[/tex]

D. [tex]\(4x^3 - 12x^2 - 2x\)[/tex]



Answer :

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]