Question

Factor the following high-degree polynomial completely. 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 \( 13x^{11} - 117x^3 \), we first look for a common factor.

1. **Factor out the greatest common factor (GCF):**
The common factor between the two terms is \( 13x^3 \):
\[
13x^{11} - 117x^3 = 13x^3(x^8 - 9)
\]

2. **Now, we need to factor \( x^8 - 9 \):**
Notice that \( 9 \) can be written as \( 3^2 \), so we can recognize this as a difference of squares:
\[
x^8 - 9 = (x^4)^2 - (3)^2 = (x^4 - 3)(x^4 + 3)
\]

3. **Putting it all together:**
We have factored the original polynomial as follows:
\[
13x^{11} - 117x^3 = 13x^3(x^4 - 3)(x^4 + 3)
\]

Thus, the completely factored form of the polynomial is:
\[
\boxed{13x^3(x^4 - 3)(x^4 + 3)}
\]