Solve {l}{ax+byay+bzaz+bx}{ay+bzaz+bxax+by}{az+bxax+byay+bz}

$$\left. \begin{array} { l } { a x + b y a y + b z a z + b x } \\ { a y + b z a z + b x a x + b y } \\ { a z + b x a x + b y a y + b z } \end{array} \right.$$

Least Common Multiple

$$lcm(ax+bx+aby^{2}+abz^{2},abx^{2}+ay+by+abz^{2},abx^{2}+aby^{2}+az+bz)$$

Solution Steps

Identify all the factors and their highest power in all expressions. Multiply the highest powers of these factors to get the least common multiple.

$$\left(abx^{2}+aby^{2}+az+bz\right)\left(abx^{2}+ay+by+abz^{2}\right)\left(ax+bx+aby^{2}+abz^{2}\right)$$

Expand the expression.

$$a^{2}b^{3}x^{5}+b^{2}a^{3}x^{5}+x^{2}a^{3}b^{3}y^{4}+a^{2}x^{2}b^{3}y^{3}+b^{2}x^{2}a^{3}y^{3}+2x^{2}y^{2}z^{2}a^{3}b^{3}+x^{2}a^{3}b^{3}z^{4}+a^{2}x^{2}b^{3}z^{3}+b^{2}x^{2}a^{3}z^{3}+axb^{3}y^{3}+bxa^{3}y^{3}+2xa^{2}b^{2}y^{3}+xa^{2}y^{2}z^{2}b^{3}+xb^{2}y^{2}z^{2}a^{3}+xyza^{3}+xyzb^{3}+3axyzb^{2}+3bxyza^{2}+axb^{3}z^{3}+bxa^{3}z^{3}+2xa^{2}b^{2}z^{3}+a^{2}b^{3}y^{5}+b^{2}a^{3}y^{5}+y^{2}a^{3}b^{3}x^{4}+a^{2}y^{2}b^{3}x^{3}+b^{2}y^{2}a^{3}x^{3}+y^{2}a^{3}b^{3}z^{4}+a^{2}y^{2}b^{3}z^{3}+b^{2}y^{2}a^{3}z^{3}+ayb^{3}x^{3}+bya^{3}x^{3}+2ya^{2}b^{2}x^{3}+ya^{2}x^{2}z^{2}b^{3}+yb^{2}x^{2}z^{2}a^{3}+ayb^{3}z^{3}+bya^{3}z^{3}+2ya^{2}b^{2}z^{3}+a^{2}b^{3}z^{5}+b^{2}a^{3}z^{5}+z^{2}a^{3}b^{3}x^{4}+a^{2}z^{2}b^{3}x^{3}+b^{2}z^{2}a^{3}x^{3}+z^{2}a^{3}b^{3}y^{4}+a^{2}z^{2}b^{3}y^{3}+b^{2}z^{2}a^{3}y^{3}+azb^{3}x^{3}+bza^{3}x^{3}+2za^{2}b^{2}x^{3}+za^{2}x^{2}y^{2}b^{3}+zb^{2}x^{2}y^{2}a^{3}+azb^{3}y^{3}+bza^{3}y^{3}+2za^{2}b^{2}y^{3}$$

Evaluate

$$ax+bx+aby^{2}+abz^{2},\ abx^{2}+ay+by+abz^{2},\ abx^{2}+aby^{2}+az+bz$$