Bài 4:
1: \(x^4y^4+4\)
\(=x^4y^4+4x^2y^2+4-4x^2y^2\)
\(=\left[\left(x^2y^2\right)^2+2\cdot x^2y^2\cdot2+2^2\right]-\left(2xy\right)^2\)
\(=\left(x^2y^2+2\right)^2-\left(2xy\right)^2\)
\(=\left(x^2y^2+2-2xy\right)\left(x^2y^2+2+2xy\right)\)
2: \(x^4y^4+64\)
\(=x^4y^4+16x^2y^2+64-16x^2y^2\)
\(=\left[\left(x^2y^2\right)^2+2\cdot x^2y^2\cdot8+8^2\right]-\left(4xy\right)^2\)
\(=\left(x^2y^2+8\right)^2-\left(4xy\right)^2\)
\(=\left(x^2y^2+8-4xy\right)\left(x^2y^2+8+4xy\right)\)
3: \(4x^4y^4+1\)
\(=4x^4y^4+1+4x^2y^2-4x^2y^2\)
\(=\left[\left(2x^2y^2\right)^2+2\cdot2x^2y^2\cdot1+1^2\right]-\left(2xy\right)^2\)
\(=\left(2x^2y^2+1\right)^2-\left(2xy\right)^2\)
\(=\left(2x^2y^2+2xy+1\right)\left(2x^2y^2-2xy+1\right)\)
4: \(32x^4+1\)
\(=\left(4\sqrt{2}\cdot x^2\right)^2+2\cdot4\sqrt{2}\cdot x^2+1-8\sqrt{2}\cdot x^2\)
\(=\left(4\sqrt{2}\cdot x^2+1\right)^2-\left(\sqrt{8\sqrt{2}}x\right)^2\)
\(=\left(4\sqrt{2}\cdot x^2+1-\sqrt{8\sqrt{2}}x\right)\left(4\sqrt{2}\cdot x^2+1+\sqrt{8\sqrt{2}}\cdot x\right)\)
5: \(x^4+4y^4\)
\(=x^4+4x^2y^2+4y^4-4x^2y^2\)
\(=\left[\left(x^2\right)^2+2\cdot x^2\cdot2y^2+\left(2y^2\right)^2\right]-\left(2xy\right)^2\)
\(=\left(x^2+2y^2\right)^2-\left(2xy\right)^2\)
\(=\left(x^2+2y^2+2xy\right)\left(x^2+2y^2-2xy\right)\)
6: \(x^7+x^2+1\)
\(=x^7+x^6+x^5-x^6-x^5-x^4+x^4+x^3+x^2-x^3+1\)
\(=x^5\left(x^2+x+1\right)-x^4\left(x^2+x+1\right)+x^2\left(x^2+x+1\right)-\left(x^3-1\right)\)
\(=\left(x^2+x+1\right)\left(x^5-x^4+x^2\right)-\left(x-1\right)\left(x^2+x+1\right)\)
\(=\left(x^2+x+1\right)\left(x^5-x^4+x^2-x+1\right)\)
7: \(x^8+x+1\)
\(=x^8+x^7+x^6-x^7-x^6-x^5+x^5+x^4+x^3-x^4-x^3-x^2+x^2+x+1\)
\(=x^6\left(x^2+x+1\right)-x^5\left(x^2+x+1\right)+x^3\left(x^2+x+1\right)-x^2\left(x^2+x+1\right)+\left(x^2+x+1\right)\)
\(=\left(x^2+x+1\right)\left(x^6-x^5+x^3-x^2+1\right)\)
8: \(x^8+x^7+1\)
\(=x^8+x^7+x^6-x^6+1\)
\(=x^6\cdot\left(x^2+x+1\right)-\left(x^6-1\right)\)
\(=x^6\left(x^2+x+1\right)-\left(x^3-1\right)\left(x^3+1\right)\)
\(=x^6\left(x^2+x+1\right)-\left(x-1\right)\left(x+1\right)\left(x^2+x+1\right)\left(x^2-x+1\right)\)
\(=\left(x^2+x+1\right)\left[x^6-\left(x^2-1\right)\left(x^2-x+1\right)\right]\)
\(=\left(x^2+x+1\right)\left[x^6-x^4+x^3-x^2+x^2-x+1\right]\)
\(=\left(x^2+x+1\right)\left(x^6-x^4+x^3-x+1\right)\)
10: \(x^{10}+x^5+1\)
\(=x^{10}+x^9+x^8-x^9-x^8-x^7+x^7+x^6+x^5-x^6+1\)
\(=x^8\left(x^2+x+1\right)-x^7\left(x^2+x+1\right)+x^5\left(x^2+x+1\right)-\left(x^6-1\right)\)
\(=\left(x^2+x+1\right)\left(x^8-x^7+x^5\right)-\left(x-1\right)\left(x+1\right)\left(x^2+x+1\right)\left(x^2-x+1\right)\)
\(=\left(x^2+x+1\right)\left[x^8-x^7+x^5-\left(x^2-1\right)\left(x^2-x+1\right)\right]\)
\(=\left(x^2+x+1\right)\left(x^8-x^7+x^5-x^4+x^3-x^2+x^2-x+1\right)\)
\(=\left(x^2+x+1\right)\left(x^8-x^7+x^5-x^4+x^3-x+1\right)\)
