Problem: (1+x2)(1−x3)\left(1+x^{2}\right)\left(1-x^{3}\right)(1+x2)(1−x3) equals
Answer Choices:
A. 1−x51-x^{5}1−x5
B. 1−x61-x^{6}1−x6
C. 1+x2−x31+x^{2}-x^{3}1+x2−x3
D. 1+x2−x3−x51+x^{2}-x^{3}-x^{5}1+x2−x3−x5
E. 1+x2−x3−x61+x^{2}-x^{3}-x^{6}1+x2−x3−x6
Solution:
(1+x2)(1−x3)=1(1+x2)−x3(1+x2)=1+x2−x3−x5.\begin{aligned} \left(1+x^{2}\right)\left(1-x^{3}\right)&=1\left(1+x^{2}\right)-x^{3}\left(1+x^{2}\right)\\ &=1+x^{2}-x^{3}-x^{5}. \end{aligned} (1+x2)(1−x3)=1(1+x2)−x3(1+x2)=1+x2−x3−x5.