Factor the following high-degree polynomial completely.

Question

13x11−117x3


(1 point)
Responses

13x3(x4−92)(x4+92)
13 x cubed left parenthesis x superscript 4 baseline minus Start Fraction 9 over 2 End Fraction right parenthesis left parenthesis x superscript 4 baseline plus Start Fraction 9 over 2 End Fraction right parenthesis

13x3(x2−3)(x2+3)(x4+3)
13 x cubed left parenthesis x squared minus 3 right parenthesis left parenthesis x squared plus 3 right parenthesis left parenthesis x superscript 4 baseline plus 3 right parenthesis

13x3(x4−3)(x4+3)
13 x cubedeft parenthesis x squared minus 8 right parenthesis left parenthesis x superscript 4 baseline plus 3 right parenthesis

13x3(x8−9)

Answer 1

To factor the polynomial \( 13x^{11} - 117x^3 \) completely, we can start by factoring out the greatest common factor (GCF) from the two terms.

1. Identify the GCF:
   • The coefficients are 13 and -117. The GCF of 13 and 117 is 13.
   • The variable part consists of \( x^{11} \) and \( x^{3} \). The GCF is \( x^{3} \).

So, we can factor out \( 13x^{3} \):

\[ 13x^{11} - 117x^{3} = 13x^{3}(x^{8} - 9) \]

2. Next, we can notice that \( x^{8} - 9 \) can be factored further. It is a difference of squares:

\[ x^{8} - 9 = (x^{4})^{2} - 3^{2} = (x^{4} - 3)(x^{4} + 3) \]

3. Putting this all together, the complete factorization of the polynomial is:

\[ 13x^{3}(x^{4} - 3)(x^{4} + 3) \]

Thus, the correct response is:

13x³(x⁴−3)(x⁴+3)