Completely factor: x 4 + 64 y 4

Question

Completely factor:       x          4        +  64      y          4

Kaiden Weeks · Accepted Answer

x          4        +  64      y          4        =      x          4        +  16      x          2            y          2        +  64      y          4        −  16      x          2            y          2        =  (      x          2        +  8      y          2            )          2        −  (  4  x  y      )          2        =  (      x          2        +  8      y          2        +  4  x  y  )  (      x          2        +  8      y          2        −  4  x  y  )

Deborah Wyatt · Accepted Answer

It can be factorized in real factors also.

x^{4} + 64 y^{4} = x^{4} + 2 * x^{2} * 8 y^{2} + 64 y^{4} - 2 * x^{2} * 8 y^{2}

= (x^{2})^{2} + 2 * x^{2} * 8 y^{2} + (8 y^{2})^{2} - 16 * x^{2} * y^{2}

= (x^{2} + 8 y^{2})^{2} - (4 x y)^{2}

= (x^{2} + 8 y^{2} - 4 x y) * (x^{2} + 8 y^{2} + 4 x y)

...........[as

A^{2} - B^{2} = (A + B) * (A - B)

]
Answer:

(x^{2} + 8 y^{2} - 4 x y) * (x^{2} + 8 y^{2} + 4 x y)

Completely factor: x^4+64y^4

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