Solve x^6+64x^3+x^3+64=

$$x ^ { 6 } + 64 x ^ { 3 } + x ^ { 3 } + 64 =$$

Factor

$$\left(x+1\right)\left(x+4\right)\left(x^{2}-x+1\right)\left(x^{2}-4x+16\right)$$

Solution Steps

Multiply and combine like terms.

$$x^{6}+65x^{3}+64$$

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

$$\left(x^{3}+64\right)\left(x^{3}+1\right)$$

Consider x^{3}+64. Rewrite x^{3}+64 as x^{3}+4^{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).

$$\left(x+4\right)\left(x^{2}-4x+16\right)$$

Consider x^{3}+1. Rewrite x^{3}+1 as x^{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).

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

Rewrite the complete factored expression. The following polynomials are not factored since they do not have any rational roots: x^{2}-x+1,x^{2}-4x+16.

$$\left(x^{2}-x+1\right)\left(x+1\right)\left(x+4\right)\left(x^{2}-4x+16\right)$$

Evaluate

$$x^{6}+65x^{3}+64$$

Solution Steps

Combine 64x^{3} and x^{3} to get 65x^{3}.

$$x^{6}+65x^{3}+64$$