Solve {l}{a^{2}b+ab^{2}-a-b*(8)*b^{2}+c^{2}-a^{2}+2bc}{4a^{2}-x^2+4xy-4y^2quadtext{(b)}a^2+b^2-2ab}

$$\left. \begin{array} { l } { a ^ { 2 } b + a b ^ { 2 } - a - b \cdot ( 8 ) \cdot b ^ { 2 } + c ^ { 2 } - a ^ { 2 } + 2 b c } \\ { 4 a ^ { 2 } - x ^ { 2 } + 4 x y - 4 y ^ { 2 } \quad \text { (b) } a ^ { 2 } + b ^ { 2 } - 2 a b } \end{array} \right.$$

Least Common Multiple

$$4a^{3}-2ba^{2}+4a^{4}-2cb^{3}-2ba^{3}+ab^{2}-8bca^{2}+4acb^{2}+2abc^{2}+8b^{5}-17ab^{4}-4ba^{4}-2b^{2}a^{3}+33a^{2}b^{3}-\left(bc\right)^{2}-\left(ax\right)^{2}-4\left(ac\right)^{2}+\left(cx\right)^{2}+\left(ab\right)^{2}+b\left(ax\right)^{2}+a\left(bx\right)^{2}+8c\left(aby\right)^{2}+4b\left(acy\right)^{2}-32b^{4}\left(ay\right)^{2}+4a^{4}\left(by\right)^{2}-4by^{2}a^{3}-4by^{2}a^{4}+4y^{2}\left(ab\right)^{3}+4axy-4xyc^{2}+4xya^{2}-8bcxy+32xyb^{3}-4bxya^{2}-4axyb^{2}-ax^{2}+2bcx^{2}-8x^{2}b^{3}$$

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.

$$ab^{2}x^{2}+ba^{2}x^{2}+c^{2}x^{2}-a^{2}x^{2}-8x^{2}b^{3}+2bcx^{2}-ax^{2}-4axyb^{2}-4bxya^{2}+32xyb^{3}-8bcxy+4xya^{2}-4xyc^{2}+4axy+4b^{2}y^{2}a^{4}+4y^{2}a^{3}b^{3}-32a^{2}y^{2}b^{4}+4ba^{2}c^{2}y^{2}+8ca^{2}b^{2}y^{2}-4by^{2}a^{4}-4by^{2}a^{3}+a^{2}b^{2}-b^{2}c^{2}+33a^{2}b^{3}-2b^{2}a^{3}-4ba^{4}-17ab^{4}+8b^{5}+2abc^{2}+4acb^{2}-8bca^{2}-4a^{2}c^{2}+ab^{2}-2ba^{3}-2cb^{3}+4a^{4}-2ba^{2}+4a^{3}$$

Expand the expression.

$$ab^{2}x^{2}+ba^{2}x^{2}+c^{2}x^{2}-8x^{2}b^{3}+2bcx^{2}-a^{2}x^{2}-ax^{2}-4axyb^{2}-4bxya^{2}+32xyb^{3}+4xya^{2}-8bcxy-4xyc^{2}+4axy+4b^{2}y^{2}a^{4}+4y^{2}a^{3}b^{3}-32a^{2}y^{2}b^{4}+4ba^{2}c^{2}y^{2}+8ca^{2}b^{2}y^{2}-4by^{2}a^{4}-4by^{2}a^{3}+a^{2}b^{2}+33a^{2}b^{3}-2b^{2}a^{3}-17ab^{4}-4ba^{4}+8b^{5}+2abc^{2}+4acb^{2}-8bca^{2}+ab^{2}-4a^{2}c^{2}-b^{2}c^{2}-2ba^{3}-2cb^{3}+4a^{4}-2ba^{2}+4a^{3}$$

Evaluate

$$ab^{2}+ba^{2}-8b^{3}+2bc+c^{2}-a^{2}-a,\ b^{2}+4a^{2}-2ab-4b\left(ay\right)^{2}+4xy-x^{2}$$