High School

What is the remainder in the synthetic division problem below?

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

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

Answer :

Certainly! Let's go through the process of solving this synthetic division problem step by step.

We are given the coefficients of a polynomial: [tex]\(1, 4, 6\)[/tex]. These correspond to the polynomial [tex]\(f(x) = 1x^2 + 4x + 6\)[/tex].

The divisor root, based on the setup of the problem, is [tex]\(-1\)[/tex]. This means we are dividing the polynomial by [tex]\(x + 1\)[/tex].

Synthetic Division Steps:

1. Set up the synthetic division: Write the coefficients of the polynomial in a row: [tex]\(1, 4, 6\)[/tex].

2. Write the root of the divisor (opposite sign of the given value): Since [tex]\(x + 1\)[/tex] is the divisor, we use [tex]\(-1\)[/tex].

3. Bring down the leading coefficient: Start by bringing down the first coefficient [tex]\(1\)[/tex] as is. It represents the new leading coefficient after division.

4. Multiply and add:
- Multiply the value you just brought down by the root of the divisor, [tex]\(-1\)[/tex], and write it under the next coefficient. So, [tex]\(1 \times (-1) = -1\)[/tex].
- Add this result to the next coefficient: [tex]\(4 + (-1) = 3\)[/tex].

5. Repeat the process:
- Multiply the result from the previous step by the root of the divisor: [tex]\(3 \times (-1) = -3\)[/tex].
- Add this to the next coefficient: [tex]\(6 + (-3) = 3\)[/tex].

The final number you get after the last addition step is the remainder of the division.

In this case, the remainder is [tex]\(3\)[/tex].

Therefore, the remainder of the synthetic division is 3, which corresponds to:
D. 3