6.21:
\(x^3-3x^2+3x-1-y^3\)
\(=\left(x-1\right)^3-y^3\)
\(=\left(x-1-y\right)\left\lbrack\left(x-1\right)^2+y\left(x-1\right)+y^2\right\rbrack\)
\(=\left(x-1-y\right)\left(x^2-2x+1+xy-y+y^2\right)\)
6.22:
\(\left(x+y\right)^3-x^3-y^3\)
\(=x^3+y^3+3xy\left(x+y\right)-x^3-y^3\)
=3xy(x+y)
7a:
\(A=x^4y^4+64\)
\(=\left(x^2y^2\right)^2+8^2\)
\(=\left(x^2y^2\right)^2+16x^2y^2+8^2-16x^2y^2\)
\(=\left(x^2y^2+8\right)^2-\left(4xy\right)^2\)
\(=\left(x^2y^2-4xy+8\right)\left(x^2y^2+4xy+8\right)\)
7b:
\(E=x^4+4y^4\)
\(=x^4+4x^2y^2+4y^4-4x^2y^2\)
\(=\left(x^2+2y^2\right)^2-\left(2xy\right)^2\)
\(=\left(x^2-2xy+2y^2\right)\left(x^2+2xy+y^2\right)\)
8.7
\(\left(x+2\right)\left(x+4\right)\left(x+6\right)\left(x+8\right)+16\)
\(=\left(x^2+10x+16\right)\left(x^2+10x+24\right)+16\)
\(=\left(x^2+10x\right)^2+40\left(x^2+10x\right)+384+16\)
\(=\left(x^2+10x\right)^2+40\left(x^2+10x\right)+400\)
\(=\left(x^2+10x+20\right)^2\)
8.4:
\(\left(x+1\right)\left(x+2\right)\left(x+3\right)\left(x+4\right)-24\)
\(=\left(x^2+5x+4\right)\left(x^2+5x+6\right)-24\)
\(=\left(x^2+5x\right)^2+10\left(x^2+5x\right)+24-24\)
\(=\left(x^2+5x\right)^2+10\left(x^2+5x\right)\)
\(=\left(x^2+5x\right)\left(x^2+5x+10\right)=x\left(x+5\right)\left(x^2+5x+10\right)\)
