To solve the polynomial division [tex]\(\left[x^3+6 x^2+5 x+2\right]:(x-1)\)[/tex], we can perform polynomial long division. Here is a step-by-step solution:
1. Set up the division:
Write the dividend [tex]\( x^3 + 6x^2 + 5x + 2 \)[/tex] inside the long division symbol and the divisor [tex]\( x - 1 \)[/tex] outside.
```
_______________
(x-1) | x^3 + 6x^2 + 5x + 2
```
2. Divide the leading term of the dividend by the leading term of the divisor:
Divide [tex]\( x^3 \)[/tex] by [tex]\( x \)[/tex] to get [tex]\( x^2 \)[/tex].
```
x^2
_______________
(x-1) | x^3 + 6x^2 + 5x + 2
```
3. Multiply the entire divisor by [tex]\( x^2 \)[/tex] and subtract:
Multiply [tex]\( (x - 1) \)[/tex] by [tex]\( x^2 \)[/tex], which results in [tex]\( x^3 - x^2 \)[/tex].
Subtract this from [tex]\( x^3 + 6x^2 + 5x + 2 \)[/tex].
```
x^2
_______________
(x-1) | x^3 + 6x^2 + 5x + 2
-(x^3 - x^2)
----------------
7x^2 + 5x + 2
```
4. Repeat the process:
Divide the leading term of the new dividend ([tex]\(7x^2\)[/tex]) by [tex]\( x \)[/tex] to get [tex]\( 7x \)[/tex].
Multiply [tex]\( (x - 1) \)[/tex] by [tex]\( 7x \)[/tex], resulting in [tex]\( 7x^2 - 7x \)[/tex].
Subtract this from [tex]\( 7x^2 + 5x + 2 \)[/tex].
```
x^2 + 7x
_______________
(x-1) | x^3 + 6x^2 + 5x + 2
-(x^3 - x^2)
----------------
7x^2 + 5x
-(7x^2 - 7x)
----------------
12x + 2
```
5. Repeat the process again:
Divide the leading term of the new dividend ([tex]\( 12x \)[/tex]) by [tex]\( x \)[/tex] to get [tex]\( 12 \)[/tex].
Multiply [tex]\( (x - 1) \)[/tex] by [tex]\( 12 \)[/tex], resulting in [tex]\( 12x - 12 \)[/tex].
Subtract this from [tex]\( 12x + 2 \)[/tex].
```
x^2 + 7x + 12
_______________
(x-1) | x^3 + 6x^2 + 5x + 2
-(x^3 - x^2)
----------------
7x^2 + 5x
-(7x^2 - 7x)
----------------
12x + 2
-(12x - 12)
----------------
14
```
6. Conclusion:
The quotient is [tex]\( x^2 + 7x + 12 \)[/tex] and the remainder is [tex]\( 14 \)[/tex].
Therefore, the result of dividing [tex]\( x^3 + 6x^2 + 5x + 2 \)[/tex] by [tex]\( x - 1 \)[/tex] is:
[tex]\[ \left(x^2 + 7x + 12\right) \text{ with a remainder of } 14 \][/tex]
Thus, the final quotient and remainder are:
[tex]\[ \boxed{x^2 + 7x + 12, 14} \][/tex]
