Solve 27x^3-1+8y^6+18xy^2

$$27 x ^ { 3 } - 1 + 8 y ^ { 6 } + 18 x y ^ { 2 }$$

Factor

$$\left(3x+2y^{2}-1\right)\left(9x^{2}-6xy^{2}+3x+4y^{4}+2y^{2}+1\right)$$

Solution Steps

Consider 27x^{3}-1+8y^{6}+18xy^{2} as a polynomial over variable x.

$$27x^{3}+18y^{2}x-1+8y^{6}$$

Find one factor of the form kx^{m}+n, where kx^{m} divides the monomial with the highest power 27x^{3} and n divides the constant factor 8y^{6}-1. One such factor is 3x+2y^{2}-1. Factor the polynomial by dividing it by this factor.

$$\left(3x+2y^{2}-1\right)\left(9x^{2}-6xy^{2}+3x+4y^{4}+2y^{2}+1\right)$$

Evaluate

$$27x^{3}+18xy^{2}+8y^{6}-1$$