Special Rules.

12. ${\displaystyle \frac{a^9-27}{a^3-3}}$.

13. ${\displaystyle \frac{a^{12}-x^6{y}^6}{a^4-x^2{y}^2}}$.

14. ${\displaystyle \frac{x^{15}-a^9{b}^9}{x^5-a^3{b}^3}}$.

15. ${\displaystyle \frac{27x^3{y}^3-z^{12}}{3xy-z^4}}$.

16. ${\displaystyle \frac{x^3{y}^3{z}^3-1}{xyz-1}}$.

17. ${\displaystyle \frac{8a^3{b}^3{c}^3-27}{2abc-3}}$.

18. ${\displaystyle \frac{1-64x^3{y}^3{z}^3}{1-4xyz}}$.

101. Sum of Two Cubes.

By performing the division, we find that $\frac{a^3+b^3}{a+b}=a^2-ab+b^2.\text{quadtext{Hence,}\}$

The sum of the cubes of two numbers is divisible by the sum of the numbers, and the quotient is the sum of the squares of the numbers minus their product.

Exercise 30.

Write by inspection the quotient of:

1. ${\displaystyle \frac{1+x^3}{1+x}}$.

2. ${\displaystyle \frac{1+8a^3}{1+2a}}$.

3. ${\displaystyle \frac{1+27c^3}{1+3c}}$.

4. ${\displaystyle \frac{8a^3+b^3}{2a+b}}$.

5. ${\displaystyle \frac{64b^3+27c^3}{4b+3c}}$.

6. ${\displaystyle \frac{27x^3+8y^3}{3x+2y}}$.

7. ${\displaystyle \frac{8x^3+125y^3}{2x+5y}}$.