Solve 3a^{4}b-6a^3b^2-9a^2b^3

$$3 a ^ { 4 } b - 6 a ^ { 3 } b ^ { 2 } - 9 a ^ { 2 } b ^ { 3 }$$

Factor

$$3b\left(a-3b\right)\left(a+b\right)a^{2}$$

Solution Steps

Factor out 3.

$$3\left(a^{4}b-2a^{3}b^{2}-3a^{2}b^{3}\right)$$

Consider a^{4}b-2a^{3}b^{2}-3a^{2}b^{3}. Factor out a^{2}b.

$$a^{2}b\left(a^{2}-2ab-3b^{2}\right)$$

Consider a^{2}-2ab-3b^{2}. Consider a^{2}-2ab-3b^{2} as a polynomial over variable a.

$$a^{2}-2ba-3b^{2}$$

Find one factor of the form a^{k}+m, where a^{k} divides the monomial with the highest power a^{2} and m divides the constant factor -3b^{2}. One such factor is a-3b. Factor the polynomial by dividing it by this factor.

$$\left(a-3b\right)\left(a+b\right)$$

Rewrite the complete factored expression.

$$3a^{2}b\left(a-3b\right)\left(a+b\right)$$

Evaluate

$$3b\left(a-3b\right)\left(a+b\right)a^{2}$$