Solve 3ab^4-6ab^3+9ab

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

Factor

$$3ab\left(b+1\right)\left(b^{2}-3b+3\right)$$

Solution Steps

Factor out 3.

$$3\left(ab^{4}-2ab^{3}+3ab\right)$$

Consider ab^{4}-2ab^{3}+3ab. Factor out ab.

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

Consider b^{3}-2b^{2}+3. By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 3 and q divides the leading coefficient 1. One such root is -1. Factor the polynomial by dividing it by b+1.

$$\left(b+1\right)\left(b^{2}-3b+3\right)$$

Rewrite the complete factored expression. Polynomial b^{2}-3b+3 is not factored since it does not have any rational roots.

$$3ab\left(b+1\right)\left(b^{2}-3b+3\right)$$

Evaluate

$$3ab\left(b+1\right)\left(b^{2}-3b+3\right)$$