Solve 2^3a^4b-a^3ab^4

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

Factor

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

Solution Steps

Factor out a^{3}b.

$$a^{3}b\left(8a-ab^{3}\right)$$

Consider 2^{3}a-ab^{3}. Factor out a.

$$a\left(8-b^{3}\right)$$

Consider 2^{3}-b^{3}. Multiply and combine like terms.

$$-b^{3}+8$$

Consider -b^{3}+8. By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 8 and q divides the leading coefficient -1. One such root is 2. Factor the polynomial by dividing it by b-2.

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

Rewrite the complete factored expression. Polynomial -b^{2}-2b-4 is not factored since it does not have any rational roots.

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

Evaluate

$$b\left(8-b^{3}\right)a^{4}$$