A(x) chia cho B(x) có số dư bằng 2. Vậy m – 5 = 2 ⇒ m = 7.

Để có phép chia hết thì số dư phải bằng 0.

Ta có: a – 5 = 0 hay a = 5.

Hay  a − 1 = 0 b + 30 = 0 ⇒ a = 1 b = − 30 .

Để \(x^4-x^3+6x^2-x+n⋮x^2-x+5\) thì

\(n-5=0\Rightarrow n=5\)

Vậy để \(x^4-x^3+6x^2-x+n⋮x^2-x+5\) thì \(n=5\)

Đặt \(f\left(x\right)=2x^3-3x^2+x+a\)

Ta có: phép chia \(f\left(x\right)\) cho \(x+2\) có dư là \(R=f\left(-2\right)\)

\(\Rightarrow f\left(-2\right)=2.\left(-2\right)^3-3.\left(-2\right)^2+\left(-2\right)+a\)

\(f\left(-2\right)=2.\left(-8\right)-3.4-2+a\)

\(f\left(-2\right)=-16-12-2+a\)

\(f\left(-2\right)=-20+a\)

Để \(f\left(x\right)\) chia hết cho \(x+2\) thì  \(R=0\) hay \(f\left(-2\right)=0\)

\(\Rightarrow-20+a=0\Leftrightarrow a=20\)

Ta có: \(\dfrac{x^4-x^3+6x^2-x+3}{x^2-x+5}\)

\(=\dfrac{x^4-x^3+5x^2+x^2-x+5-2}{x^2-x+5}\)

\(=x^2+1-\dfrac{2}{x^2-x+5}\)

\(x^4-x^3+6x^2-x+a=x^2\left(x^2-x+5\right)+x^2-x+a\)

Do \(x^2\left(x^2-x+5\right)\) chia hết \(x^2-x+5\)

\(\Rightarrow x^2-x+a\) chia hết \(x^2-x+5\)

\(\Rightarrow a=5\)

b: \(\Leftrightarrow3n^3+n^2+9n^2+3n-3n-1-4⋮3n+1\)

\(\Leftrightarrow3n+1\in\left\{1;-1;2;-2;4;-4\right\}\)

\(\Leftrightarrow n\in\left\{0;-1;1\right\}\)