To factor the expression \( 13x^{11} - 117x^3 \) completely, we can follow these steps:
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Factor out the greatest common factor (GCF): In this expression, both terms share a common factor of \( 13x^3 \).
\[ 13x^{11} - 117x^3 = 13x^3 (x^8 - 9) \]
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Factor the difference of squares: The expression \( x^8 - 9 \) can be factored further, as it is a difference of squares:
\[ x^8 - 9 = (x^4)^2 - 3^2 = (x^4 - 3)(x^4 + 3) \]
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Combine the factors: Substituting back, we have:
\[ 13x^{11} - 117x^3 = 13x^3 (x^4 - 3)(x^4 + 3) \]
Since \( x^4 - 3 \) and \( x^4 + 3 \) do not factor further using real coefficients, we conclude our factorization here.
Thus, the completely factored form of the expression is:
\[ \boxed{13x^3 (x^4 - 3)(x^4 + 3)} \]