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Answer :
The polynomial P(x) = 625x^4 - 16 can be factored completely by recognizing it as a difference of squares and breaking it down to (25x^2 + 4)(5x + 2)(5x - 2).
Here in the above question we have to factor the polynomial P(x) = 625x^4 - 16 completely, we first recognize that this is a difference of squares.
A difference of squares can be factored as A^2 - B^2 = (A + B)(A - B).
Here, we can see that 625x^4 is the square of 25x^2 and 16 is the square of 4.
Therefore, the polynomial can be rewritten as: (25x^2 + 4)(25x^2 - 4)
We can further factor the second term, 25x^2 - 4, as it is also a difference of squares: (25x^2 - 4) = (5x + 2)(5x - 2)
Hence, the polynomial P(x) = 625x^4 - 16 is factored completely as: (25x^2 + 4)(5x + 2)(5x - 2)
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