To perform synthetic division for the polynomial [tex]\( x^3 - 10x^2 + 12x + 3 \)[/tex] divided by [tex]\( x - 2 \)[/tex], follow these steps:
1. Set up the synthetic division table:
```
2 | 1 -10 12 3
|
```
The coefficients of the polynomial [tex]\( x^3 - 10x^2 + 12x + 3 \)[/tex] are [tex]\( 1, -10, 12, \)[/tex] and [tex]\( 3 \)[/tex]. The number [tex]\( 2 \)[/tex] comes from the divisor [tex]\( x - 2 \)[/tex].
2. Bring down the leading coefficient:
```
2 | 1 -10 12 3
|
-----------------
1
```
The leading coefficient is [tex]\( 1 \)[/tex], and we bring it down below the line.
3. Multiply and add:
- Multiply the number below the line (1) by the divisor (2), and write the result under the next coefficient (-10).
```
2 | 1 -10 12 3
| 2
-----------------
1 -8
```
- Add the numbers in the second column: [tex]\(-10 + 2 = -8\)[/tex].
- Multiply -8 (the result just obtained) by the divisor (2), and write the result under the next coefficient (12).
```
2 | 1 -10 12 3
| 2 -16
-----------------
1 -8 -4
```
- Add the numbers in the third column: [tex]\( 12 + (-16) = -4 \)[/tex].
- Multiply -4 by the divisor (2), and write the result under the next coefficient (3).
```
2 | 1 -10 12 3
| 2 -16 -8
-----------------
1 -8 -4 -5
```
- Add the numbers in the fourth column: [tex]\( 3 + (-8) = -5 \)[/tex].
4. Interpret the results:
The bottom row, excluding the final number, gives the coefficients of the quotient polynomial. The final number is the remainder.
So, the quotient is:
[tex]\( 1x^2 - 8x - 4 \)[/tex]
And the remainder is:
[tex]\( -5 \)[/tex]
Therefore, when [tex]\( x^3 - 10x^2 + 12x + 3 \)[/tex] is divided by [tex]\( x-2 \)[/tex] using synthetic division, the quotient is [tex]\( x^2 - 8x - 4 \)[/tex] and the remainder is [tex]\( -5 \)[/tex].
So, we can write:
[tex]\[ \frac{x^3 - 10x^2 + 12x + 3}{x-2} = x^2 - 8x - 4 \text{ remainder } -5 \][/tex]
