Solve (x-2y^3)^2-(2y^3-x)^2

$$( x - 2 y ^ { 3 } ) ^ { 2 } - ( 2 y ^ { 3 } - x ) ^ { 2 }$$

Evaluate

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Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-2y^{3}\right)^{2}.

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

To raise a power to another power, multiply the exponents. Multiply 3 and 2 to get 6.

$$x^{2}-4xy^{3}+4y^{6}-\left(2y^{3}-x\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(2y^{3}-x\right)^{2}.

$$x^{2}-4xy^{3}+4y^{6}-\left(4\left(y^{3}\right)^{2}-4y^{3}x+x^{2}\right)$$

To raise a power to another power, multiply the exponents. Multiply 3 and 2 to get 6.

$$x^{2}-4xy^{3}+4y^{6}-\left(4y^{6}-4y^{3}x+x^{2}\right)$$

Combine x^{2}-4xy^{3}+4y^{6} and -\left(4y^{6}-4y^{3}x+x^{2}\right) to get 0.

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Factor

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Solution Steps

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

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Consider 2x-4y^{3}. Factor out 2.

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

Rewrite the complete factored expression. Simplify.

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