Solve x^7+16x^3-8x^5=

$$x ^ { 7 } + 16 x ^ { 3 } - 8 x ^ { 5 } =$$

Factor

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

Solution Steps

Factor out x^{3}.

$$x^{3}\left(x^{4}+16-8x^{2}\right)$$

Consider x^{4}+16-8x^{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 16. One such factor is x^{2}-4. Factor the polynomial by dividing it by this factor.

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

Consider x^{2}-4. Rewrite x^{2}-4 as x^{2}-2^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(x-2\right)\left(x+2\right)$$

Rewrite the complete factored expression.

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

Evaluate

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