dễ mà bạn xin 20 phút làm ra giấy nhé :))
a) \(\left(x^4+6x^3+9x^2\right)+2x^2+6x+1\)
\(\left(x^2+3x\right)^2+2\left(x^2+3x\right)+1\)
\(\left(x^2+3x+1\right)^2\)
b) \(x^4+x^3+x^2+x+1\)
câu b, chúa sẽ c/m x ko tồn tại , và nó là 1 đa thức bất khả Q . trong R
vì lớp 8 chưa học đến số phức
\(x^4+x^3=-x^2-x-1\)
\(x^4+x^3+\frac{1}{4}x^2=\left(\frac{1}{4}x^2-x^2\right)-x-1\)
\(\left(x^2+\frac{1}{2}x\right)^2=-\frac{3}{4}x^2-x-1\)
\(4\left(x^2+\frac{1}{2}x\right)^2=-3x^2-4x-4\)
\(\Delta`=\left(-2\right)^2-\left(-4\right).\left(-3\right)=4-12< 0\)
denta < 0 x vô nghiệm
vậy đa thức trên ko thể phân tích và nó là 1 đa thức bất khả Q
c) ,
\(\left(6x^4-12x^3\right)+\left(17x^3-34x^2\right)-\left(4x^2-8x\right)-\left(3x-6\right)\)
\(6x^3\left(x-2\right)+17x^2\left(x-2\right)-4x\left(x-2\right)-3\left(x-2\right)\)
\(\left(x-2\right)\left(6x^3+17x^2-4x-3\right)\)
\(\left(x-2\right)\left\{\left(6x^3+18x^2\right)-\left(x^2+3x\right)-\left(x+3\right)\right\}\)
\(\left(x-2\right)\left\{6x^2\left(x+3\right)-x\left(x+3\right)-\left(x+3\right)\right\}\)
\(\left(x-2\right)\left(x+3\right)\left(6x^2-x-1\right)\)
\(\left(x-2\right)\left(x+3\right)\left\{\left(6x^2+\frac{6}{3}x\right)-\left(\frac{9}{3}x+\frac{9}{9}\right)\right\}\)
\(\left(x-2\right)\left(x+3\right)\left\{6x\left(x+\frac{1}{3}\right)-\frac{9}{3}\left(x+\frac{1}{3}\right)\right\}\)
\(\left(X-2\right)\left(X+3\right)\left(X+\frac{1}{3}\right)\left(6x-1\right)\)
d)
\(\left(x^4-x^3\right)+\left(6x^3-6x^2\right)-\left(6x^2-6x\right)-\left(x-1\right)\)
\(x^3\left(x-1\right)+6x^2\left(x-1\right)-6x\left(x-1\right)-\left(x-1\right)\)
\(\left(x-1\right)\left(x^3+6x^2-6x-1\right)\)
\(\left(x-1\right)\left\{\left(x^3-x^2\right)+\left(7x^2-7x\right)+\left(x-1\right)\right\}\)
\(\left(x-1\right)^2\left(x^2+7x+1\right)\)
\(\Delta=49-4=45\)
\(x1,2=\frac{-7+\sqrt{45}}{2},\frac{-7-\sqrt{45}}{2}\)
\(\left(x-1\right)^2\left(x-\frac{7+\sqrt{45}}{2}\right)\left(x-\frac{7-\sqrt{45}}{2}\right)\)
