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Find the quotient [tex]\(\frac{10x^6 + 18x^5 - 6x}{2x^2}\)[/tex].

A. [tex]\(8x^4 + 16x^3 - 4x^2\)[/tex]

B. [tex]\(5x^2 + 9x^3 - 3x\)[/tex]

C. [tex]\(5x^3 + 9x^3 + 3x^2\)[/tex]

D. [tex]\(5x^4 + 9x^3 - 3x^2\)[/tex]

Answer :

To find the quotient of the given expression, we'll divide the polynomial in the numerator by the polynomial in the divisor term by term.

### Step-by-Step Solution

1. Expression Setup:
We start with the polynomial expression to divide:

[tex]\[
\frac{10x^6 + 18x^5 - 6x}{2x^2}
\][/tex]

2. Divide Each Term:
We will divide each term in the numerator by the term in the divisor, [tex]\(2x^2\)[/tex].

- Divide [tex]\(10x^6\)[/tex] by [tex]\(2x^2\)[/tex]:
[tex]\[
\frac{10x^6}{2x^2} = 5x^4
\][/tex]

- Divide [tex]\(18x^5\)[/tex] by [tex]\(2x^2\)[/tex]:
[tex]\[
\frac{18x^5}{2x^2} = 9x^3
\][/tex]

- Divide [tex]\(-6x\)[/tex] by [tex]\(2x^2\)[/tex]:
[tex]\[
\frac{-6x}{2x^2} = -3x^{-1} \text{ (We ignore this term in our solution because it represents a negative power)}
\][/tex]

3. Quotient:
Combine the results of the division:

- The quotient is:
[tex]\[
5x^4 + 9x^3
\][/tex]

Therefore, the quotient of the division [tex]\(\frac{10x^6 + 18x^5 - 6x}{2x^2}\)[/tex] is [tex]\(5x^4 + 9x^3\)[/tex].

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