Solve x^2-28x^3:7x^3

$$x ^ { 2 } - 28 x ^ { 3 } : 7 x ^ { 3 }$$

Evaluate

$$x^{2}-4x^{6}$$

Solution Steps

Divide 28x^{3} by 7 to get 4x^{3}.

$$x^{2}-4x^{3}x^{3}$$

To multiply powers of the same base, add their exponents. Add 3 and 3 to get 6.

$$x^{2}-4x^{6}$$

Factor

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

Solution Steps

Factor out x^{2}.

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

Consider 1-4x^{4}. Rewrite 1-4x^{4} as 1^{2}-\left(2x^{2}\right)^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

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

Reorder the terms.

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

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

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