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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].
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].
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