1.
a) \(x^2-3x+2=\left(x^2-x\right)+\left(-2x+2\right)=x\left(x-1\right)-2\left(x-1\right)=\left(x-2\right)\left(x-1\right)\)
b) \(4x^2+81=\left(4x^2+36x+81\right)-36x=\left(2x+9\right)^2-36x=\left(2x+9-\sqrt{36x}\right)\left(2x+9+\sqrt{36x}\right)\)
c) \(x^5+x^4+1=\left(x^5-x^2\right)+\left(x^4+x^2+1\right)\)
\(=x^2\left(x^3-1\right)+\left(x^2+1\right)-x^2\)
\(=x^2\left(x-1\right)\left(x^2+x+1\right)+\left(x^2+x+1\right)\left(x^2-x+1\right)=\left(x^2+x+1\right)\left[x^2\left(x-1\right)+x^2-x+1\right]\)d)A = \(x\left(x+4\right)\left(x+6\right)\left(x+10\right)+128\)
\(=\left(x^2+10x\right)\left(x^2+10x+24\right)+128\)
Đặt \(t=x^2+10x\)
Khi đó \(A=t\left(t+24\right)+128\)
\(=t^2+24t+128\)
\(=\left(t+12\right)^2-16\)
\(=\left(t+8\right)\left(t+16\right)\)
\(=\left(x^2+10x+8\right)\left(x^2+10x+16\right)\)
\(=\left[\left(x+5\right)^2-17\right]\left[\left(x+5\right)^2-9\right]\)
\(=\left(x+5+\sqrt{17}\right)\left(x+5-\sqrt{17}\right)\left(x+2\right)\left(x+8\right)\)
2.
\(1^2-2^2+3^2-4^2+....+99^2-100^2\)
\(=\left(1+2\right)\left(1-2\right)+\left(3-4\right)\left(3+4\right)+...+\left(99-100\right)\left(99+100\right)\)
\(=-3-7-11-...-199\)
\(=-\left(3+7+11+...+199\right)\)
\(=-5050\)
Bước cuối bạn tự tính nhé <33
