Question

Drag and drop the expressions into the boxes to correctly complete the proof of the polynomial identity.(x2 + y2)2 + 2x?y– y4 = x(x² + 4y?)(x2 + y2)2 + 2x²y2 – y4 = x(+ 4y?)+2x²y2 – y4 = x2 (x2 + 4y?)x² (x² + 47²)= x2 (x2 + 4y2)x² (x² + 47²) x² – 2x²y² + y x² + yt x² + 4x²72 x + 2x²,2x² + 2x²y² + yt

Answer 1

x^4 + y^4 + 2x^2 y^2

x^4 + 4x^2y^2

x^2 (x^2 + 4y^2 )

Explanation:

Expanding the the expression gives

`\begin{gathered} (x^2+y^2)^4=(x^2)^2+(y^2)^2+2(x^2)(y^2) \\ =\boxed{x^4+y^4+2x^2y^2\text{.}} \end{gathered}`

Simplifying the Left-hand side gives

`\begin{gathered} x^4+y^4+2x^2y^2+2x^2y^2-y^4 \\ =\boxed{x^4+4x^2y^2\text{.}} \end{gathered}`

Finally, factoring out x^2 from the left-hand side gives

`x^4+4x^2y^2=\boxed{x^2\mleft(x^2+4y^2\mright)\text{.}}`