Solve {l}{x^{3}+x^{5}-x^{7}}{9a^2-12ab+15a^3b^2-24ab^3}

$$\left. \begin{array} { l } { x ^ { 3 } + x ^ { 5 } - x ^ { 7 } } \\ { 9 a ^ { 2 } - 12 a b + 15 a ^ { 3 } b ^ { 2 } - 24 a b ^ { 3 } } \end{array} \right.$$

Least Common Multiple

$$3ax^{3}\left(-5x^{4}\left(ab\right)^{2}+8b^{3}x^{4}+4bx^{4}-3ax^{4}-8x^{2}b^{3}+3ax^{2}-4bx^{2}+5\left(abx\right)^{2}+5\left(ab\right)^{2}-8b^{3}+3a-4b\right)$$

Solution Steps

Factor the expressions that are not already factored.

$$x^{3}\left(1+x^{2}-x^{4}\right)=-\left(x^{2}-\left(-\frac{1}{2}\sqrt{5}+\frac{1}{2}\right)\right)\left(x^{2}-\left(\frac{1}{2}\sqrt{5}+\frac{1}{2}\right)\right)x^{3}$$

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

$$3ax^{3}\left(-5a^{2}b^{2}x^{4}+8b^{3}x^{4}+4bx^{4}-3ax^{4}+5a^{2}b^{2}x^{2}-8x^{2}b^{3}+3ax^{2}-4bx^{2}+5a^{2}b^{2}-8b^{3}+3a-4b\right)$$

Expand the expression.

$$-15b^{2}a^{3}x^{7}+24ab^{3}x^{7}+12abx^{7}-9a^{2}x^{7}+15b^{2}a^{3}x^{5}-24ab^{3}x^{5}+9a^{2}x^{5}-12abx^{5}+15b^{2}a^{3}x^{3}-24ab^{3}x^{3}+9a^{2}x^{3}-12abx^{3}$$

Evaluate

$$x^{3}\left(1+x^{2}-x^{4}\right),\ 3a\left(5\left(ab\right)^{2}-8b^{3}+3a-4b\right)$$