Solve {r}{2(2a^{2}-3b^{2}+4c^{2})+3(b^{2}-a^{2}+c^{2})+4(a^{2}-b^{2}-c^{2})}{4y(3y^{2}+5y-7)+2(y^3-4y^2+5)}{+3(6y-5y^2-3y^3-2)}

$$\left. \begin{array} { r } { 2 ( 2 a ^ { 2 } - 3 b ^ { 2 } + 4 c ^ { 2 } ) + 3 ( b ^ { 2 } - a ^ { 2 } + c ^ { 2 } ) + 4 ( a ^ { 2 } - b ^ { 2 } - c ^ { 2 } ) } \\ { 4 y ( 3 y ^ { 2 } + 5 y - 7 ) + 2 ( y ^ { 3 } - 4 y ^ { 2 } + 5 ) } \\ { + 3 ( 6 y - 5 y ^ { 2 } - 3 y ^ { 3 } - 2 ) } \end{array} \right.$$

Least Common Multiple

$$6\left(5a^{2}+7c^{2}-7b^{2}\right)\left(21y^{6}+53y^{5}-54y^{4}-77y^{3}+121y^{2}-58y+10\right)$$

Solution Steps

Factor the expressions that are not already factored.

$$10-28y+12y^{2}+14y^{3}=2\left(7y^{3}+6y^{2}-14y+5\right) 18y-15y^{2}-9y^{3}-6=3\left(-3y^{3}-5y^{2}+6y-2\right)$$

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

$$6\left(5a^{2}+7c^{2}-7b^{2}\right)\left(21y^{6}+53y^{5}-54y^{4}-77y^{3}+121y^{2}-58y+10\right)$$

Expand the expression.

$$630a^{2}y^{6}+882c^{2}y^{6}-882b^{2}y^{6}+1590a^{2}y^{5}+2226c^{2}y^{5}-2226b^{2}y^{5}+2268b^{2}y^{4}-2268c^{2}y^{4}-1620a^{2}y^{4}+3234b^{2}y^{3}-3234c^{2}y^{3}-2310a^{2}y^{3}+3630a^{2}y^{2}+5082c^{2}y^{2}-5082b^{2}y^{2}+2436yb^{2}-2436yc^{2}-1740ya^{2}-420b^{2}+300a^{2}+420c^{2}$$

Evaluate

$$5a^{2}+7c^{2}-7b^{2},\ 14y^{3}+12y^{2}-28y+10,\ -9y^{3}-15y^{2}+18y-6$$