To factor the polynomials:
f(x) = x^2 - 8
We can use the difference of squares formula: a^2 - b^2 = (a + b)(a - b). In this case, a = x and b = sqrt(8) = 2sqrt(2).
f(x) = x^2 - (2sqrt(2))^2
= (x + 2sqrt(2))(x - 2sqrt(2))
Therefore, the factored form of f(x) is (x + 2sqrt(2))(x - 2sqrt(2)).
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f(x) = 25x^2 -12
We can factor out a common factor of 1 from each term:
f(x) = 1(25x^2) - 1(12)
Next, we can use the difference of squares formula again, with a = 5x and b = sqrt(12) = 2sqrt(3):
f(x) = (5x)^2 - (2sqrt(3))^2
= (5x + 2sqrt(3))(5x - 2sqrt(3))
Therefore, the factored form of f(x) is (5x + 2sqrt(3))(5x - 2sqrt(3)).