\(A=x^4-6x^3+27x^2-54x+32\)

\(=x^4-5x^3+22x^2-32x-x^3+5x^2-22x+32\)

\(=x\left(x^3-5x^2+22x-32\right)-\left(x^3-5x^2+22x-32\right)\)

\(=\left(x-1\right)\left(x^3-5x^2+22x-32\right)\)

\(=\left(x-1\right)\left(x^3-3x^2+16x-2x^2+6x-32\right)\)

\(=\left(x-1\right)\left[x\left(x^2-3x+16\right)-2\left(x^2-3x+16\right)\right]\)

\(=\left(x-1\right)\left(x-2\right)\left(x^2-3x+16\right)\)

Vì \(x\in Z\)=> x-1;x-2 là 2 số nguyên liên tiếp => \(\left(x-1\right)\left(x-2\right)⋮2\)

\(\Rightarrow A=\left(x-1\right)\left(x-2\right)\left(x^2-3x+16\right)⋮2\) hay A là số chẵn (đpcm)

\(A=x^4-6x^3+27x^2-54x+32\)

\(=x^4-x^3-5x^3+5x^2+22x^2-22x-32x+32\)

\(=\left(x-1\right)\left(x^3-5x^2+22x-32\right)\)

\(=\left(x-1\right)\left[x^2\left(x-2\right)-3x\left(x-2\right)+16\left(x-2\right)\right]\)

\(=\left(x-1\right)\left(x-2\right)\left(x^2-3x+16\right)\)

Vì \(\left(x-1\right)\left(x-2\right)⋮2\) nên A là số chẵn với mọi x thuộc Z

 f(x) = x⁴ + 6x³ +11x² + 6x 

\(=x^4+x^3+5x^3+5x^2+6x^2+6x\)

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

\(=x^3\left(x+1\right)+5x^2\left(x+1\right)+6x\left(x+1\right)\)

\(=\left(x+1\right)\left(x^3+5x^2+6x\right)\)

\(=x\left(x+1\right)\left(x^2+5x+6\right)\)

\(=x\left(x+1\right)\left[x^2+2x+3x+6\right]\)

\(=x\left(x+1\right)\left[\left(x^2+2x\right)+\left(3x+6\right)\right]\)

\(=x\left(x+1\right)\left[x\left(x+2\right)+3\left(x+2\right)\right]\)

\(=x\left(x+1\right)\left(x+2\right)\left(x+3\right)\)

b)Ta có

\(f\left(x\right)+1=x\left(x+1\right)\left(x+2\right)\left(x+3\right)+1\)

\(=\left[x\left(x+3\right)\right].\left[\left(x+1\right)\left(x+2\right)\right]+1\)

\(=\left(x^2+3x\right).\left(x^2 +3x+2\right)+1\)

\(=\left(x^2+3x+1-1\right).\left(x^2+3x+1+1\right)+1\)

\(=\left[\left(x^2+3x+1\right)-1\right].\left[\left(x^2+3x+1\right)+1\right]+1\)

\(=\left(x^2+3x+1\right)^2-1+1=\left(x^2+3x+1\right)^2\)

Vậy với mọi x nguyên thì f(x) + 1 luôn có giá trị là 1 số chính phương 

ý D nhé

a/ \(x^3-5x^2+6x+3=\left(x-2\right)\left(x^2-3x\right)+3.\)( Dùng phép chia đa thức)

Để A chia hết cho x-2 thì 3 phải chia hết cho x-2 => x-2 là ước của 3

=> x-2={3-; -1; 1; 3} => x={-1; 1; 3; 5}

b/ Chia F(x) cho x-1

\(f\left(x\right)=\left(x-1\right)\left(x^2-5x+6\right)\)

Giải phương trình bậc 2 \(x^2-5x+6=0\) để tìm nghiệm còn lại

\(A=x^4-6x^3+27x^2-54x+32\)

\(=x^4-2x^3-4x^3+8x^2+19x^2-38x-16x+32\)

\(=x^3\left(x-2\right)-4x^2\left(x-2\right)+19x\left(x-2\right)-16\left(x-2\right)\)

\(=\left(x^3-4x^2+19x-16\right)\left(x-2\right)\)

A= x^4 - 6x^3 + 27x^2 - 54x + 32

A= x^4 - 3x^3 + 2x^2 - 3x^3 + 9x^2 - 6x + 16x^2 - 48x + 32

A= x^2(x^2 - 3x + 2) - 3x(x^2 - 3x + 2) + 16(x^2 - 3x + 2)

A= (x^2 - 3x + 2) (x^2 - 3x + 16)

Chúc bạn học giỏi nhé!

Tiếp theo cái bài tớ vừa đăng ý!

A=(x^2 - 3x + 2)(x^2 - 3x + 16)

A=(x^2 - x - 2x + 2)(x^2 - 3x + 16)

A=[x(x - 1) - 2(x - 1)](x^2 - 3x + 16)

A=(x - 1)(x - 2)(x^2 - 3x + 16)

x^2 - 3x + 16 vô nghiệm vì:

x^2-3x+16=x^2 - 2.3/2 + 9/4+55/4=(x-3/2)^2+55/4>0 với mọi x

(vì (x-3/2)^2 >=0)

\(x^2-y^2+4x+4\)

\(=\left(x+2\right)^2-y^2\)

\(=\left(x+2+y\right)\left(x+2-y\right)\)

\(4x^2-y^2+8\left(y-2\right)\)

\(=4x^2-\left(y^2-8y+16\right)\)

\(=4x^2-\left(y-4\right)^2\)

\(=\left(2x+y-4\right)\left(2x-y+4\right)\)