To factor the polynomial \( 16x^8 - 49x^2 \), we first look for a common factor in both terms.
Notice that both terms contain \( x^2 \), so we can factor that out:
\[ 16x^8 - 49x^2 = x^2(16x^6 - 49) \]
Next, we have a difference of squares in the expression \( 16x^6 - 49 \), which can be written as \( (4x^3)^2 - 7^2 \). The difference of squares can be factored using the identity \( a^2 - b^2 = (a - b)(a + b) \).
Thus, we can factor \( 16x^6 - 49 \) as follows:
\[ 16x^6 - 49 = (4x^3 - 7)(4x^3 + 7) \]
Putting it all together, we have:
\[ 16x^8 - 49x^2 = x^2(4x^3 - 7)(4x^3 + 7) \]
So, the completely factored form of the polynomial is:
\[ \boxed{x^2(4x^3 - 7)(4x^3 + 7)} \]
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