Question

Factor the following high-degree polynomial completely.13x11−117x3 (1 point)Responses13x3(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 parenthesis13x3(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 parenthesis13x3(x4−3)(x4+3)13 x cubedeft parenthesis x squared minus 8 right parenthesis left parenthesis x superscript 4 baseline plus 3 right parenthesis13x3(x8−9)

Answer 1

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

1. Factor out the GCD (greatest common divisor): The GCD of \( 13x^{11} \) and \( 117x^3 \) is \( 13x^3 \).

   So we factor \( 13x^3 \) out of the polynomial:

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

2. Factor \( x^8 - 9 \): The expression \( x^8 - 9 \) is a difference of squares:

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

3. Write the complete factorization: Now we can substitute this back into our expression:

   \[ 13x^{11} - 117x^3 = 13x^3((x^4 - 3)(x^4 + 3)) \]

Thus, the completely factored form of the polynomial is:

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

So the correct response is: \[ \boxed{13x^3(x^4 - 3)(x^4 + 3)} \]