Solve 27x+x^5-12x^3

$$27 x + x ^ { 5 } - 12 x ^ { 3 }$$

Factor

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

Solution Steps

Factor out x.

$$x\left(27+x^{4}-12x^{2}\right)$$

Consider 27+x^{4}-12x^{2}. Find one factor of the form x^{k}+m, where x^{k} divides the monomial with the highest power x^{4} and m divides the constant factor 27. One such factor is x^{2}-9. Factor the polynomial by dividing it by this factor.

$$\left(x^{2}-9\right)\left(x^{2}-3\right)$$

Consider x^{2}-9. Rewrite x^{2}-9 as x^{2}-3^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(x-3\right)\left(x+3\right)$$

Rewrite the complete factored expression. Polynomial x^{2}-3 is not factored since it does not have any rational roots.

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

Evaluate

$$x\left(x^{4}-12x^{2}+27\right)$$