Solve 2x^5y-20x^3y+18xy=

$$2 x ^ { 5 } y - 20 x ^ { 3 } y + 18 x y =$$

Factor

$$2xy\left(x-3\right)\left(x-1\right)\left(x+1\right)\left(x+3\right)$$

Solution Steps

Factor out 2.

$$2\left(x^{5}y-10x^{3}y+9xy\right)$$

Consider x^{5}y-10x^{3}y+9xy. Factor out xy.

$$xy\left(x^{4}-10x^{2}+9\right)$$

Consider x^{4}-10x^{2}+9. Find one factor of the form x^{k}+m, where x^{k} divides the monomial with the highest power x^{4} and m divides the constant factor 9. One such factor is x^{2}-9. Factor the polynomial by dividing it by this factor.

$$\left(x^{2}-9\right)\left(x^{2}-1\right)$$

Consider x^{2}-9. Rewrite x^{2}-9 as x^{2}-3^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(x-3\right)\left(x+3\right)$$

Consider x^{2}-1. Rewrite x^{2}-1 as x^{2}-1^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(x-1\right)\left(x+1\right)$$

Rewrite the complete factored expression.

$$2xy\left(x-3\right)\left(x+3\right)\left(x-1\right)\left(x+1\right)$$

Evaluate

$$2xy\left(x^{2}-9\right)\left(x^{2}-1\right)$$