Solve 8x^6-15x^3-2

$$8 x ^ { 6 } - 15 x ^ { 3 } - 2$$

Factor

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

Solution Steps

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

$$\left(8x^{3}+1\right)\left(x^{3}-2\right)$$

Consider 8x^{3}+1. Rewrite 8x^{3}+1 as \left(2x\right)^{3}+1^{3}. The sum of cubes can be factored using the rule: a^{3}+b^{3}=\left(a+b\right)\left(a^{2}-ab+b^{2}\right).

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

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

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

Evaluate

$$8x^{6}-15x^{3}-2$$