Solve 8p^3+12p^2q-18pq^2-27q^3

$$8 p ^ { 3 } + 12 p ^ { 2 } q - 18 p q ^ { 2 } - 27 q ^ { 3 }$$

Factor

$$\left(2p-3q\right)\left(2p+3q\right)^{2}$$

Solution Steps

Consider 8p^{3}+12p^{2}q-18pq^{2}-27q^{3} as a polynomial over variable p.

$$8p^{3}+12qp^{2}-18q^{2}p-27q^{3}$$

Find one factor of the form kp^{m}+n, where kp^{m} divides the monomial with the highest power 8p^{3} and n divides the constant factor -27q^{3}. One such factor is 2p-3q. Factor the polynomial by dividing it by this factor.

$$\left(2p-3q\right)\left(4p^{2}+12pq+9q^{2}\right)$$

Consider 4p^{2}+12pq+9q^{2}. Use the perfect square formula, a^{2}+2ab+b^{2}=\left(a+b\right)^{2}, where a=2p and b=3q.

$$\left(2p+3q\right)^{2}$$

Rewrite the complete factored expression.

$$\left(2p-3q\right)\left(2p+3q\right)^{2}$$

Evaluate

$$\left(2p-3q\right)\left(2p+3q\right)^{2}$$