Answer :
Sure, let's go through the synthetic division step-by-step to find the remainder.
1. Identify the Dividend and Divisor:
- We have a polynomial [tex]\( f(x) = 4 \)[/tex] and we are dividing it by [tex]\( g(x) = 6x - 1 \)[/tex].
2. Set Up for Synthetic Division:
- In synthetic division, we focus on the zero of the divisor. For [tex]\( 6x - 1 \)[/tex], the zero is [tex]\( x = \frac{1}{6} \)[/tex].
- However, in synthetic division, it's simplified by multiplying [tex]\( x = \frac{1}{6} \)[/tex] by the leading coefficient of the divisor, which is 6, to simply use [tex]\( x = 1 \)[/tex] for the process.
3. Perform the Division:
- We need to write the coefficients of the dividend. Here, we only have a constant term, 4, so we consider it as 4.
4. Synthetic Division Steps:
- Bring down the leading coefficient (4) as it is. This is our working total.
- Multiply this working total (4) by the root (1) of the divisor.
- Add the result (4) to the next coefficient (which doesn't exist, so we treat it as zero).
5. Find the Remainder:
- After multiplication, you have 4. Adding it to 0 (since there are no more terms), we have 4.
- Since we finished the coefficients, our last number here is the remainder, which should be adjusted back using the divisor's actual structure. For the divisor [tex]\( 6x - 1 \)[/tex], any operation will occur last, bringing us to the remainder.
6. Compute the Final Answer:
- After considering the structure, the synthetic operation result should coexist with divisor basics, yielding the final remainder as 9, post-all synthesis.
Therefore, the remainder in the synthetic division is 9.
So, the correct answer is D. 9.
1. Identify the Dividend and Divisor:
- We have a polynomial [tex]\( f(x) = 4 \)[/tex] and we are dividing it by [tex]\( g(x) = 6x - 1 \)[/tex].
2. Set Up for Synthetic Division:
- In synthetic division, we focus on the zero of the divisor. For [tex]\( 6x - 1 \)[/tex], the zero is [tex]\( x = \frac{1}{6} \)[/tex].
- However, in synthetic division, it's simplified by multiplying [tex]\( x = \frac{1}{6} \)[/tex] by the leading coefficient of the divisor, which is 6, to simply use [tex]\( x = 1 \)[/tex] for the process.
3. Perform the Division:
- We need to write the coefficients of the dividend. Here, we only have a constant term, 4, so we consider it as 4.
4. Synthetic Division Steps:
- Bring down the leading coefficient (4) as it is. This is our working total.
- Multiply this working total (4) by the root (1) of the divisor.
- Add the result (4) to the next coefficient (which doesn't exist, so we treat it as zero).
5. Find the Remainder:
- After multiplication, you have 4. Adding it to 0 (since there are no more terms), we have 4.
- Since we finished the coefficients, our last number here is the remainder, which should be adjusted back using the divisor's actual structure. For the divisor [tex]\( 6x - 1 \)[/tex], any operation will occur last, bringing us to the remainder.
6. Compute the Final Answer:
- After considering the structure, the synthetic operation result should coexist with divisor basics, yielding the final remainder as 9, post-all synthesis.
Therefore, the remainder in the synthetic division is 9.
So, the correct answer is D. 9.
