Answer :
To find the remainder using synthetic division for the problem [tex]$1 \longdiv { 4 6 - 3 }$[/tex], we can follow these steps:
1. Identify the Coefficients and Divisor:
The polynomial given is represented by its coefficients: [tex]\(4, 6, -3\)[/tex]. This means the polynomial is [tex]\(4x^2 + 6x - 3\)[/tex].
Since the divisor is [tex]\(x - 1\)[/tex], we will use the value [tex]\(1\)[/tex] for synthetic division.
2. Set Up the Synthetic Division:
We write down the coefficients of the polynomial: [tex]\(4, 6, -3\)[/tex]. Then, place the root of the divisor, which is [tex]\(1\)[/tex], to the left.
3. Perform Synthetic Division:
- Bring down the first coefficient (4), which will be used as the initial value.
- Multiply this value by the root (1) and add it to the next coefficient.
- [tex]\(4 \times 1 = 4\)[/tex], then add to [tex]\(6\)[/tex], getting [tex]\(10\)[/tex].
- Repeat the process: multiply [tex]\(10\)[/tex] by [tex]\(1\)[/tex] and then add to [tex]\(-3\)[/tex].
- [tex]\(10 \times 1 = 10\)[/tex], then add to [tex]\(-3\)[/tex], getting [tex]\(7\)[/tex].
4. Determine the Remainder:
The last number you obtain in the synthetic division process is the remainder. In this case, the remainder is [tex]\(7\)[/tex].
Thus, the remainder when [tex]\((4x^2 + 6x - 3)\)[/tex] is divided by [tex]\((x - 1)\)[/tex] is [tex]\(\boxed{7}\)[/tex].
1. Identify the Coefficients and Divisor:
The polynomial given is represented by its coefficients: [tex]\(4, 6, -3\)[/tex]. This means the polynomial is [tex]\(4x^2 + 6x - 3\)[/tex].
Since the divisor is [tex]\(x - 1\)[/tex], we will use the value [tex]\(1\)[/tex] for synthetic division.
2. Set Up the Synthetic Division:
We write down the coefficients of the polynomial: [tex]\(4, 6, -3\)[/tex]. Then, place the root of the divisor, which is [tex]\(1\)[/tex], to the left.
3. Perform Synthetic Division:
- Bring down the first coefficient (4), which will be used as the initial value.
- Multiply this value by the root (1) and add it to the next coefficient.
- [tex]\(4 \times 1 = 4\)[/tex], then add to [tex]\(6\)[/tex], getting [tex]\(10\)[/tex].
- Repeat the process: multiply [tex]\(10\)[/tex] by [tex]\(1\)[/tex] and then add to [tex]\(-3\)[/tex].
- [tex]\(10 \times 1 = 10\)[/tex], then add to [tex]\(-3\)[/tex], getting [tex]\(7\)[/tex].
4. Determine the Remainder:
The last number you obtain in the synthetic division process is the remainder. In this case, the remainder is [tex]\(7\)[/tex].
Thus, the remainder when [tex]\((4x^2 + 6x - 3)\)[/tex] is divided by [tex]\((x - 1)\)[/tex] is [tex]\(\boxed{7}\)[/tex].
