Solve y^6-16y^3+64

$$y ^ { 6 } - 16 y ^ { 3 } + 64$$

Factor

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

Solution Steps

Find one factor of the form y^{k}+m, where y^{k} divides the monomial with the highest power y^{6} and m divides the constant factor 64. One such factor is y^{3}-8. Factor the polynomial by dividing it by this factor.

$$\left(y^{3}-8\right)\left(y^{3}-8\right)$$

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

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

Rewrite the complete factored expression. Polynomial y^{2}+2y+4 is not factored since it does not have any rational roots.

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

Evaluate

$$\left(y^{3}-8\right)^{2}$$