Solve 48x^9+144x^6-72x^3-6

$$48 { x }^{ 9 } +144 { x }^{ 6 } -72 { x }^{ 3 } -6$$

Factor

$$6\left(2x^{3}-1\right)\left(4x^{6}+14x^{3}+1\right)$$

Solution Steps

Factor out 6.

$$6\left(8x^{9}+24x^{6}-12x^{3}-1\right)$$

Consider 8x^{9}+24x^{6}-12x^{3}-1. Find one factor of the form kx^{m}+n, where kx^{m} divides the monomial with the highest power 8x^{9} and n divides the constant factor -1. One such factor is 2x^{3}-1. Factor the polynomial by dividing it by this factor.

$$\left(2x^{3}-1\right)\left(4x^{6}+14x^{3}+1\right)$$

Rewrite the complete factored expression. The following polynomials are not factored since they do not have any rational roots: 2x^{3}-1,4x^{6}+14x^{3}+1.

$$6\left(2x^{3}-1\right)\left(4x^{6}+14x^{3}+1\right)$$

Evaluate

$$48x^{9}+144x^{6}-72x^{3}-6$$