High School

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

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

Answer :

To find the quotient of the expression [tex]\(-12x^9 + 3x^7 + 24x^6\)[/tex] when divided by [tex]\(6x\)[/tex], we divide each term of the polynomial by [tex]\(6x\)[/tex] separately. Let's go through this step by step:

1. Divide the first term [tex]\(-12x^9\)[/tex] by [tex]\(6x\)[/tex]:
- Coefficients: [tex]\(-12\)[/tex] divided by [tex]\(6\)[/tex] gives [tex]\(-2\)[/tex].
- Exponents: [tex]\(x^9\)[/tex] divided by [tex]\(x\)[/tex] means subtracting the exponents: [tex]\(9 - 1 = 8\)[/tex].
- Result: [tex]\(-2x^8\)[/tex].

2. Divide the second term [tex]\(3x^7\)[/tex] by [tex]\(6x\)[/tex]:
- Coefficients: [tex]\(3\)[/tex] divided by [tex]\(6\)[/tex] gives [tex]\(\frac{1}{2}\)[/tex].
- Exponents: [tex]\(x^7\)[/tex] divided by [tex]\(x\)[/tex] means subtracting the exponents: [tex]\(7 - 1 = 6\)[/tex].
- Result: [tex]\(\frac{1}{2}x^6\)[/tex].

3. Divide the third term [tex]\(24x^6\)[/tex] by [tex]\(6x\)[/tex]:
- Coefficients: [tex]\(24\)[/tex] divided by [tex]\(6\)[/tex] gives [tex]\(4\)[/tex].
- Exponents: [tex]\(x^6\)[/tex] divided by [tex]\(x\)[/tex] means subtracting the exponents: [tex]\(6 - 1 = 5\)[/tex].
- Result: [tex]\(4x^5\)[/tex].

Combining these results, the quotient is:

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

Therefore, the correct answer is:
D. [tex]\(-2x^8 + \frac{1}{2}x^6 + 4x^5\)[/tex].