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Câu 1:
\(a^2+2ab+b^2-ac-bc\)
\(=\left(a+b\right)^2-c\left(a+b\right)\)
\(=\left(a+b\right)\left(a+b-c\right)\)
Câu 2:
\(5x^2-5y^2-10x+10y\)
\(=5\left(x-y\right)\left(x+y\right)-10\left(x-y\right)\)
\(=\left(x-y\right)\left(5x+5y-10\right)\)
\(=5\left(x-y\right)\left(x+y-2\right)\)
Câu 3:
\(3x^2-6xy+3y^2-12z^2\)
\(=3\left[\left(x-y\right)^2-4z^2\right]\)
\(=3\left(x-y-2z\right)\left(x-y+2z\right)\)
Câu 4:
\(x^4+x^3+x^2-1\)
\(=x^3\left(x+1\right)+\left(x-1\right)\left(x+1\right)\)
\(=\left(x+1\right)\left(x^3+x-1\right)\)
Câu 5:
\(x^3-3x^2+3x-1-y^3\)
\(=\left(x-1\right)^3-y^3\)
\(=\left(x-1-y\right)\left[\left(x-1\right)^2+\left(x-1\right)y+y^2\right]\)
\(=\left(x-y-1\right)\left(x^2-2x+1+xy-y+y^2\right)\)
Câu 6:
\(x^4-x^2+2x-1\)
\(=x^4-\left(x-1\right)^2\)
\(=\left(x^2-x+1\right)\left(x^2+x-1\right)\)
Câu 7:
\(\left(x+y\right)^3-x^3-y^3\)
\(=\left(x+y\right)^3-\left[\left(x+y\right)^3-3xy\left(x+y\right)\right]\)
\(=3xy\left(x+y\right)\)
\(\left(x+2\right)\left(x+3\right)\left(x+4\right)\left(x+5\right)-24\)
\(=\left(x^2+7x+10\right)\left(x^2+7x+12\right)-24\)
Đặt \(x^2+7x+10=a\)ta có:
\(a\left(a+2\right)-24\)
\(=a^2+2a+1-25\)
\(=\left(a+1\right)^2-25\)
\(=\left(a-4\right)\left(a+6\right)\)
Thay trở lại ta được: \(\left(x^2+7x+6\right)\left(x^2+7x+16\right)\)
\(\left(x+2\right)\left(x+3\right)\left(x+4\right)\left(x+5\right)-24\)
\(=\left(x^2+7x+10\right)\left(x^2+7x+12\right)-24\)
Đặt \(x^2+7x+10=a\) ta có:
\(a\left(a+2\right)-24\)
\(=a^2+2a+1-25\)
\(=\left(a+1\right)^2-25\)
\(=\left(a-4\right)\left(a+6\right)\)
Thay trở lại ta được: \(\left(x^2+7x+6\right)\left(x^2+7x+16\right)\)
\(\left(x+2\right)\left(x+3\right)\left(x+4\right)\left(x+5\right)-24\)
\(=\left(x^2+7x+10\right)\left(x^2+7x+12\right)-24\)
Đặt \(t=x^2+7x+11\)
đến đây biến đổi theo t rồi thay trở lại
a) Ta có: \(x^5+x-1\)
\(=x^5-x^4+x^3+x^4-x^3+x^2-x^2+x-1\)
\(=x^3\left(x^2-x+1\right)+x^2\left(x^2-x+1\right)-\left(x^2-x+1\right)\)
\(=\left(x^2-x+1\right)\left(x^3+x^2-1\right)\)
b) Ta có: \(x^5+x^4+1\)
\(=x^5+x^4+x^3-x^3+1\)
\(=x^3\left(x^2+x+1\right)-\left(x^3-1\right)\)
\(=x^3\left(x^2+x+1\right)-\left(x-1\right)\left(x^2+x+1\right)\)
\(=\left(x^3+x+1\right)\left(x^3-x+1\right)\)
c) Ta có: \(x^7+x^2+1\)
\(=x^7+x^6+x^5-x^6-x^5-x^4+x^4+x^3+x^2-x^3-x^2-x+x^2+x+1\)
\(=x^5\left(x^2+x+1\right)-x^4\left(x^2+x+1\right)+x^2\left(x^2+x+1\right)-x\left(x^2+x+1\right)+\left(x^2+x+1\right)\)
\(=\left(x^2+x+1\right)\left(x^5-x^4+x^2-x+1\right)\)
d) Ta có: \(x^7+x^5+1\)
\(=x^7+x^6+x^5-x^6+1\)
\(=x^5\left(x^2+x+1\right)-\left(x^6-1\right)\)
\(=x^5\left(x^2+x+1\right)-\left(x^3+1\right)\left(x^3-1\right)\)
\(=x^5\left(x^2+x+1\right)-\left(x-1\right)\left(x^2+x+1\right)\left(x^3+1\right)\)
\(=\left(x^2+x+1\right)\left[x^5-\left(x-1\right)\left(x^3+1\right)\right]\)
\(=\left(x^2+x+1\right)\left[x^5-\left(x^4+x-x^3-1\right)\right]\)
\(=\left(x^2+x+1\right)\left(x^5-x^4-x+x^3+1\right)\)
e) Ta có: \(x^8+x^7+1\)
\(=x^8+x^7+x^6-\left(x^6-1\right)\)
\(=x^6\left(x^2+x+1\right)-\left(x-1\right)\left(x^2+x+1\right)\left(x^3+1\right)\)
\(=\left(x^2+x+1\right)\left[x^6-\left(x-1\right)\left(x^3+1\right)\right]\)
\(=\left(x^2+x+1\right)\left[x^6-\left(x^4+x-x^3-1\right)\right]\)
\(=\left(x^2+x+1\right)\left(x^6-x^4-x+x^3+1\right)\)
f) Ta có: \(x^{10}+x^5+1\)
\(=\left(x^{10}-x\right)+\left(x^5-x^2\right)+\left(x^2+x+1\right)\)
\(=x\left(x^9-1\right)+x^2\left(x-1\right)\left(x^2+x+1\right)+\left(x^2+x+1\right)\)
\(=x\left(x-1\right)\left(x^2+x+1\right)\left(x^6+x^3+1\right)+x^2\left(x-1\right)\left(x^2+x+1\right)+\left(x^2+x+1\right)\)
\(=\left(x^2+x+1\right)\left[\left(x^2-x\right)\left(x^6+x^3+1\right)+x^3-x^2+1\right]\)
\(=\left(x^2+x+1\right)\left[x^8+x^5+x^2-x^7-x^4-x+x^3-x^2+1\right]\)
\(=\left(x^2+x+1\right)\left(x^8-x^7+x^5-x^4+x^3-x+1\right)\)