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Select the correct answer.

What is the quotient when [tex]$(-12x^9 + 3x^7 + 24x^6)$[/tex] is divided by [tex]$6x$[/tex]?

A. [tex]$-2x^8 + 2x^6 + 4x^5$[/tex]

B. [tex]$-2x^8 + \frac{1}{2}x^6 + 4x^5$[/tex]

C. [tex]$2x^8 + \frac{1}{2}x^6 + 4x^5$[/tex]

D. [tex]$2x^9 + 2x^7 + 4x^6$[/tex]

Answer :

Sure! Let's solve the problem step by step.

We need to find the quotient when the polynomial [tex]\((-12x^9 + 3x^7 + 24x^6)\)[/tex] is divided by [tex]\(6x\)[/tex].

To do this, we'll divide each term of the polynomial by [tex]\(6x\)[/tex].

1. First Term: [tex]\(-12x^9 \div 6x\)[/tex]
[tex]\[
\frac{-12x^9}{6x} = -2x^{9-1} = -2x^8
\][/tex]

2. Second Term: [tex]\(3x^7 \div 6x\)[/tex]
[tex]\[
\frac{3x^7}{6x} = \frac{3}{6}x^{7-1} = \frac{1}{2}x^6
\][/tex]

3. Third Term: [tex]\(24x^6 \div 6x\)[/tex]
[tex]\[
\frac{24x^6}{6x} = 4x^{6-1} = 4x^5
\][/tex]

Now, let's write the quotient from the simplified terms:

[tex]\[
-2x^8 + \frac{1}{2}x^6 + 4x^5
\][/tex]

The correct answer is option B: [tex]\(-2x^8 + \frac{1}{2}x^6 + 4x^5\)[/tex].

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