Đáp án:

 

Giải thích các bước giải:

`1)`

$\left.\begin{array}{l} x^4 - 9x^3 + 21x^2 + x + k \\ x^4 - 2x^3 \\ \hline \ \ \ \ \ - 7x^3 + 21x^2 \\ \ \ \ \ \ - 7x^3 + 14x^2 \\ \hline \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 7x^2 + x \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 7x^2 - 14x \\ \hline \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 15x + k \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 15x - 30 \\ \hline \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ k + 30 \end{array}\right|\!\!\!\;\!\!\begin{array}{l} \dfrac{\ \ x - 2 \ \ }{\ \ x^3 - 7x^2 + 7x + 15 \ \ } \\ \\ \\ \\ \\ \\ \\ \\ \end{array}$

Để `P(x) vdots G(x)` thì số dư phải bằng `0`:

`k + 30 = 0`

`k = -30`

`2)`

$\left.\begin{array}{l} x^4 - 10x^3 + 27x^2 + 8x + k \\ x^4 + 2x^3 \\ \hline \ \ \ \ \ - 12x^3 + 27x^2 \\ \ \ \ \ \ - 12x^3 - 24x^2 \\ \hline \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 51x^2 + 8x \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 51x^2 + 102x \\ \hline \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ - 94x + k \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ - 94x - 188 \\ \hline \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ k + 188 \end{array}\right|\!\!\!\;\!\!\begin{array}{l} \dfrac{\ \ x + 2 \ \ }{\ \ x^3 - 12x^2 + 51x - 94 \ \ } \\ \\ \\ \\ \\ \\ \\ \\ \end{array}$

Để `P(x) vdots G(x)` thì số dư phải bằng `0`:

`k + 188 = 0`

`k = -188`

`3)`

$\left.\begin{array}{l} x^4 - 19x^3 + 25x^2 - 6x + k \\ x^4 - 3x^3 \\ \hline \ \ \ \ \ - 16x^3 + 25x^2 \\ \ \ \ \ \ - 16x^3 + 48x^2 \\ \hline \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ - 23x^2 - 6x \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ - 23x^2 + 69x \\ \hline \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ - 75x + k \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ - 75x + 225 \\ \hline \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ k - 225 \end{array}\right|\!\!\!\;\!\!\begin{array}{l} \dfrac{\ \ x - 3 \ \ }{\ \ x^3 - 16x^2 - 23x - 75 \ \ } \\ \\ \\ \\ \\ \\ \\ \\ \end{array}$

Để `P(x) vdots G(x)` thì số dư phải bằng `0`:

`k - 225 = 0`

`k = 225`