To factor the polynomial \( 16x^8 - 49x^2 \), we can start by looking for a common factor and then apply the difference of squares pattern.
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Factor out the common term: Both terms have a common factor of \( x^2 \):
\[ 16x^8 - 49x^2 = x^2(16x^6 - 49) \]
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Recognize the difference of squares: The expression \( 16x^6 - 49 \) can be recognized as a difference of squares. Recall that \( a^2 - b^2 = (a - b)(a + b) \).
Here, we can identify:
- \( a = 4x^3 \) (since \( (4x^3)^2 = 16x^6 \))
- \( b = 7 \) (since \( 7^2 = 49 \))
Thus, we can rewrite \( 16x^6 - 49 \):
\[ 16x^6 - 49 = (4x^3 - 7)(4x^3 + 7) \]
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Combine everything: Now substitute back to get the complete factorization:
\[ 16x^8 - 49x^2 = x^2(4x^3 - 7)(4x^3 + 7) \]
So, the fully factored form of the polynomial \( 16x^8 - 49x^2 \) is:
\[ x^2(4x^3 - 7)(4x^3 + 7) \]