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------------------------------------------------ What is the remainder in the synthetic division problem below?

[tex]\[ 1 \div \begin{array}{c} 4 \\ 6 \\ -1 \end{array} \][/tex]

A. 7
B. 5
C. 3
D. 9

Sure! Let's go through the steps of synthetic division to find the remainder when dividing the polynomial by [tex]\( x - 1 \)[/tex].

The polynomial given is represented by the coefficients [tex]\([4, 6, -1]\)[/tex], which means it is [tex]\( 4x^2 + 6x - 1 \)[/tex]. We are dividing by [tex]\( x - 1 \)[/tex], where the divisor [tex]\( x - c \)[/tex] implies [tex]\( c = 1 \)[/tex].

Here's a step-by-step breakdown:

1. Set Up the Division: Write down the coefficients: 4, 6, and -1.

2. Bring Down the Leading Coefficient: Start by bringing down the first coefficient, which is 4.

3. Multiply and Add:
- Multiply the number brought down (4) by [tex]\( c \)[/tex] (which is 1). So, [tex]\( 4 \times 1 = 4 \)[/tex].
- Add this result to the next coefficient (6): [tex]\( 6 + 4 = 10 \)[/tex].

4. Repeat the Process:
- Take the new number (10) and multiply it by [tex]\( c \)[/tex] (1): [tex]\( 10 \times 1 = 10 \)[/tex].
- Add this result to the next coefficient (-1): [tex]\( -1 + 10 = 9 \)[/tex].

5. Result:
- The number obtained after the last addition is the remainder: 9.

So, the remainder of the synthetic division is 9.

Therefore, the correct answer is:
D. 9