Solve E=(x-y)(x+y)(x^2+y^2)+y^4

$$E = ( x - y ) ( x + y ) ( x ^ { 2 } + y ^ { 2 } ) + y ^ { 4 }$$

Solve for E

$$E=x^{4}$$

Solution Steps

Use the distributive property to multiply x-y by x+y and combine like terms.

$$E=\left(x^{2}-y^{2}\right)\left(x^{2}+y^{2}\right)+y^{4}$$

Consider \left(x^{2}-y^{2}\right)\left(x^{2}+y^{2}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$E=\left(x^{2}\right)^{2}-\left(y^{2}\right)^{2}+y^{4}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$E=x^{4}-\left(y^{2}\right)^{2}+y^{4}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$E=x^{4}-y^{4}+y^{4}$$

Combine -y^{4} and y^{4} to get 0.

$$E=x^{4}$$

Solve for x (complex solution)

$$x=i\sqrt[4]{E}$$

$$x=\sqrt[4]{E}$$

$$x=-\sqrt[4]{E}$$

$$x=-i\sqrt[4]{E}$$

Solve for x

$$x=\sqrt[4]{E}$$

$$x=-\sqrt[4]{E}\text{, }E\geq 0$$