High School

What is the remainder in the synthetic division problem below?

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

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

Answer :

Sure! Let's solve the synthetic division problem step by step.

We are given the polynomial represented by the coefficients [tex]\(4\)[/tex], [tex]\(6\)[/tex], and [tex]\(-1\)[/tex]. We need to divide this by [tex]\((x - 1)\)[/tex], since our divisor is [tex]\(1\)[/tex].

Synthetic division is a simpler way to divide a polynomial by a binomial of the form [tex]\(x - c\)[/tex].

Here is how the process goes:

1. Set Up the Problem:
- Write down the coefficients: [tex]\(4, 6, -1\)[/tex].

2. Synthetic Division Steps:
- Bring down the first coefficient [tex]\(4\)[/tex] as is.
- Multiply this number [tex]\(4\)[/tex] by the divisor [tex]\(1\)[/tex] (since we divide by [tex]\(x - 1\)[/tex], [tex]\(c = 1\)[/tex]), which gives us [tex]\(4\)[/tex].
- Add this product to the next coefficient: [tex]\(6 + 4 = 10\)[/tex].
- Multiply the result [tex]\(10\)[/tex] by the divisor [tex]\(1\)[/tex]: [tex]\(10 \times 1 = 10\)[/tex].
- Add this result to the next coefficient: [tex]\(-1 + 10 = 9\)[/tex].

3. Conclusion:
- The last result we get after adding is the remainder of the division.

The remainder is thus [tex]\(9\)[/tex].

So, the correct answer is B. 9.