\(a)x^4-9x^3+21x^2+x+a\\ =\left(x^4-2x^3\right)+\left(-7x^3+14x^2\right)+\left(7x^2-14x\right)+\left(15x-30\right)+\left(a+30\right)\\ =x^3\left(x-2\right)-7x^2\left(x-2\right)+7x\left(x-2\right)+15\left(x-2\right)+\left(a+30\right)\\ =\left(x-2\right)\left(x^3-7x^2+7x+15\right)+\left(a+30\right)\)
Để `x^4-9x^3+21x^2+x+a` ⋮ x - 2 thì a + 30 = 0 <=> a = -30
\(b)3x^4-7x^3+11x^2+x-a\\ =\left(3x^4-12x^3\right)+\left(5x^3-20x^2\right)+\left(31x^2-124x\right)+\left(125x-500\right)+\left(500-a\right)\\ =3x^3\left(x-4\right)+5x^2\left(x-4\right)+31x\left(x-4\right)+125\left(x-4\right)+\left(500-a\right)\\ =\left(x-4\right)\left(3x^3+5x^2+31x+125\right)+\left(500-a\right)\)
Để `3x^4-7x^3+11x^2+x-a` ⋮ x - 4 thì 500 - a = 0 <=> a = 500
\(c)x^4-x^3+6x^2-x+a\\ =\left(x^4-x^3+5x^2\right)+\left(x^2-x+5\right)+\left(a-5\right)\\ =x^2\left(x^2-x+5\right)+\left(x^2-x+5\right)+\left(a-5\right)\\ =\left(x^2+1\right)\left(x^2-x+5\right)+\left(a-5\right)\)
Để `x^4-x^3+6x^2-x+a` ⋮ `x^2-x+5` thì a - 5 = 0 <=> a = 5
