Solve 216pq-p^7q

$$216 p q - p ^ { 7 } q$$

Factor

$$pq\left(p^{2}-6\right)\left(-p^{4}-6p^{2}-36\right)$$

Solution Steps

Factor out pq.

$$pq\left(216-p^{6}\right)$$

Consider 216-p^{6}. Find one factor of the form kp^{m}+n, where kp^{m} divides the monomial with the highest power -p^{6} and n divides the constant factor 216. One such factor is p^{2}-6. Factor the polynomial by dividing it by this factor.

$$\left(p^{2}-6\right)\left(-p^{4}-6p^{2}-36\right)$$

Rewrite the complete factored expression. The following polynomials are not factored since they do not have any rational roots: -p^{4}-6p^{2}-36,p^{2}-6.

$$pq\left(p^{2}-6\right)\left(-p^{4}-6p^{2}-36\right)$$

Evaluate

$$pq\left(216-p^{6}\right)$$