High School

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

[tex]
\[
1 \quad \overline{4} \quad 6 \quad -1
\]
[/tex]

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

Answer :

To find the remainder when using synthetic division, follow these steps:

1. Identify the root: Since the divisor is given as [tex]\( x - 4 \)[/tex], the root is 4.

2. Write down the coefficients: For the polynomial described, the coefficients are 1, 6, and -1.

3. Begin synthetic division:
- Start with the first coefficient, which is 1.
- Multiply this number by the root (4) and add the result to the next coefficient.

4. Perform the calculations step-by-step:
- Start with the first coefficient: 1.
- Multiply 1 by the root (4): [tex]\( 1 \times 4 = 4 \)[/tex].
- Add the result to the second coefficient (6): [tex]\( 4 + 6 = 10 \)[/tex].
- Now multiply 10 by the root (4): [tex]\( 10 \times 4 = 40 \)[/tex].
- Add this result to the last coefficient (-1): [tex]\( 40 + (-1) = 39 \)[/tex].

5. Conclusion: The remainder is the final number obtained after all the additions, which is 39.

So, the remainder when dividing the polynomial by [tex]\( x - 4 \)[/tex] is 39.