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Divide the polynomial [tex]$12x^7 + 48x^6 - 54x^4 - 6x$[/tex] by 6. What is the new polynomial?

A. [tex]$10x^{13} - 9x^5$[/tex]
B. [tex]$2x^7 + 8x^6 - 9x^4 - x$[/tex]
C. [tex]$12x + 48 - 54x^{-1} - 6x^{-5}$[/tex]
D. [tex]$2x^7 + 48x^6 - 54x^4 - x$[/tex]

Answer :

Sure! Let's go through the steps to divide the polynomial [tex]\(12x^7 + 48x^6 - 54x^4 - 6x\)[/tex] by 6.

### Step 1: Break down each term
You need to divide each term of the polynomial by 6 individually:

1. [tex]\(12x^7\)[/tex] divided by 6:
[tex]\[
\frac{12x^7}{6} = 2x^7
\][/tex]

2. [tex]\(48x^6\)[/tex] divided by 6:
[tex]\[
\frac{48x^6}{6} = 8x^6
\][/tex]

3. [tex]\(-54x^4\)[/tex] divided by 6:
[tex]\[
\frac{-54x^4}{6} = -9x^4
\][/tex]

4. [tex]\(-6x\)[/tex] divided by 6:
[tex]\[
\frac{-6x}{6} = -x
\][/tex]

### Step 2: Write the new polynomial
Now, combine all the terms you obtained from dividing each one by 6:
[tex]\[
2x^7 + 8x^6 - 9x^4 - x
\][/tex]

This is the new polynomial after division. The correct choice from the given options is:
[tex]\[
2x^7 + 8x^6 - 9x^4 - x
\][/tex]

I hope this helps you understand how to perform polynomial division! If you have any more questions, feel free to ask.

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Rewritten by : Barada