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

[tex]2 \longdiv { 2 - 4 - 9 1 }[/tex]

A. -17
B. 5
C. -1
D. -19

Answer :

Sure, let's go through the synthetic division step-by-step:

We have the polynomial [tex]\( P(x) = 2x^3 - 4x^2 - 9x + 1 \)[/tex] and we are dividing it by [tex]\( x - 2 \)[/tex]. The divisor, [tex]\( x - 2 \)[/tex], means we'll use the value 2 for our synthetic division.

1. Set up the coefficients: The coefficients of the polynomial are 2, -4, -9, and 1.

2. Carry down the leading coefficient: We start by bringing down the first coefficient, which is 2.

3. Multiply and add:
- Multiply the number brought down (2) by the divisor (2), which gives 4.
- Add this result (4) to the next coefficient (-4). The sum is 0.

4. Repeat the process:
- Multiply the result (0) by the divisor (2), resulting in 0.
- Add it to the next coefficient (-9). The sum is -9.
- Multiply -9 by the divisor (2), resulting in -18.
- Add this to the last coefficient (1), giving a final sum of -17.

The remainder of this synthetic division is -17.

So, the correct answer is A. -17.