Answer :
To find the quotient when the expression [tex]\((-12x^9 + 3x^7 + 24x^6)\)[/tex] is divided by [tex]\(6x\)[/tex], we need to divide each term in the expression by [tex]\(6x\)[/tex]. Here's how you can do it step by step:
1. Divide the first term:
- The first term is [tex]\(-12x^9\)[/tex].
- Divide [tex]\(-12\)[/tex] by [tex]\(6\)[/tex] to get [tex]\(-2\)[/tex].
- Divide [tex]\(x^9\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(x^{9-1} = x^8\)[/tex].
- So, [tex]\((-12x^9) \div (6x) = -2x^8\)[/tex].
2. Divide the second term:
- The second term is [tex]\(3x^7\)[/tex].
- Divide [tex]\(3\)[/tex] by [tex]\(6\)[/tex] to get [tex]\(\frac{1}{2}\)[/tex].
- Divide [tex]\(x^7\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(x^{7-1} = x^6\)[/tex].
- So, [tex]\((3x^7) \div (6x) = \frac{1}{2}x^6\)[/tex].
3. Divide the third term:
- The third term is [tex]\(24x^6\)[/tex].
- Divide [tex]\(24\)[/tex] by [tex]\(6\)[/tex] to get [tex]\(4\)[/tex].
- Divide [tex]\(x^6\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(x^{6-1} = x^5\)[/tex].
- So, [tex]\((24x^6) \div (6x) = 4x^5\)[/tex].
Now, combine the results of each division:
- The quotient of the entire polynomial is [tex]\(-2x^8 + \frac{1}{2}x^6 + 4x^5\)[/tex].
Based on this, the correct answer is:
D. [tex]\(-2x^8 + \frac{1}{2}x^6 + 4x^5\)[/tex].
1. Divide the first term:
- The first term is [tex]\(-12x^9\)[/tex].
- Divide [tex]\(-12\)[/tex] by [tex]\(6\)[/tex] to get [tex]\(-2\)[/tex].
- Divide [tex]\(x^9\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(x^{9-1} = x^8\)[/tex].
- So, [tex]\((-12x^9) \div (6x) = -2x^8\)[/tex].
2. Divide the second term:
- The second term is [tex]\(3x^7\)[/tex].
- Divide [tex]\(3\)[/tex] by [tex]\(6\)[/tex] to get [tex]\(\frac{1}{2}\)[/tex].
- Divide [tex]\(x^7\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(x^{7-1} = x^6\)[/tex].
- So, [tex]\((3x^7) \div (6x) = \frac{1}{2}x^6\)[/tex].
3. Divide the third term:
- The third term is [tex]\(24x^6\)[/tex].
- Divide [tex]\(24\)[/tex] by [tex]\(6\)[/tex] to get [tex]\(4\)[/tex].
- Divide [tex]\(x^6\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(x^{6-1} = x^5\)[/tex].
- So, [tex]\((24x^6) \div (6x) = 4x^5\)[/tex].
Now, combine the results of each division:
- The quotient of the entire polynomial is [tex]\(-2x^8 + \frac{1}{2}x^6 + 4x^5\)[/tex].
Based on this, the correct answer is:
D. [tex]\(-2x^8 + \frac{1}{2}x^6 + 4x^5\)[/tex].
