Compound Integral Expressions.

4. Divide $10a^2{b}^2-20b^4-17a^{3}b+6a^4+a{b}^3$ by $2a^2-4b^2-3ab$.

Arrange according to descending powers of $a$. $$\begin{gathered} r*4cr|l6a^4\&-\&17a^{3}b\&+\&10a^2{b}^2\&+\&a{b}^3\&-\&20b^4\&2a^2-3ab-4b^2 \\ \mathrm{\backslash cline}10-106a^4\&-\&9a^{3}b\&-\&12a^2{b}^2\&\&\&\&\&3a^2-4ab+5b^2 \\ \mathrm{\backslash cline}1-9\&-\&8a^{3}b\&+\&22a^2{b}^2\&+\&a{b}^3\&-\&\mathrm{\backslash multicolumn}1@r20b^4 \\ \&-\&8a^{3}b\&+\&12a^2{b}^2\&+\&16a{b}^3 \\ \mathrm{\backslash cline}2-9\&\&\&\&10a^2{b}^2\&-\&15a{b}^3\&-\&\mathrm{\backslash multicolumn}1@r20b^4 \\ \&\&\&\&10a^2{b}^2\&-\&15a{b}^3\&-\&\mathrm{\backslash multicolumn}1@r20b^4 \\ \mathrm{\backslash cline}5-9 \end{gathered}$$

5. Divide $5x^3-3x^4-4x^2+1+x$ by $1+2x-3x^2$.

Arrange according to ascending powers of $x$. $$\begin{gathered} r*4cr|l*2cr1\&+\&x\&-\&4x^2\&+\&5x^3\&-\&3x^4\&1\&+\&2x\&-\&3x^2 \\ \mathrm{\backslash cline}10-141\&+\&2x\&-\&3x^2\&\&\&\&\&1\&-\&x\&+\&x^2 \\ \mathrm{\backslash cline}1-9\&-\&x\&-\&x^2\&+\&5x^3\&-\&\mathrm{\backslash multicolumn}1@r3x^4 \\ \&-\&x\&-\&2x^2\&+\&3x^3 \\ \mathrm{\backslash cline}2-9\&\&\&\&x^2\&+\&2x^3\&-\&\mathrm{\backslash multicolumn}1@r3x^4 \\ \&\&\&\&x^2\&+\&2x^3\&-\&\mathrm{\backslash multicolumn}1@r3x^4 \end{gathered}$$

6. Divide $a^3+b^3+c^3-3abc$ by $a+b+c$.

Arrange according to descending powers of $a$. $$\begin{gathered} r*4crclcr|l{a}^3\&\&\&\&\&-\&3abc\&\&\&+\&b^3\&+\&c^3\&a+b+c \\ \mathrm{\backslash cline}14-14a^3\&+\&a^{2}b\&+\&a^{2}c\&\&\&\&\&\&\&\&\&[t]a^2\&-ab-ac \\ \&+b^2-bc+c^2 \\ \mathrm{\backslash cline}1-13\&-\&a^{2}b\&-\&a^{2}c\&-\&3abc\&\&\&+\&b^3\&+\&\mathrm{\backslash multicolumn}1@r{c}^3 \\ \&-\&a^{2}b\&-\&a{b}^2\&-\&abc \\ \mathrm{\backslash cline}2-13\&-\&a^{2}c\&+\&a{b}^2\&-\&2abc\&\&\&+\&b^3\&+\&\mathrm{\backslash multicolumn}1@r{c}^3 \\ \&-\&a^{2}c\&\&\&-\&abc\&-\&a{c}^2 \\ \mathrm{\backslash cline}2-13\&\&\&\&a{b}^2\&-\&abc\&+\&a{c}^2\&+\&b^3\&+\&\mathrm{\backslash multicolumn}1@r{c}^3 \\ \&\&\&\&a{b}^2\&\&\&\&\&+\&b^3\&+\&\mathrm{\backslash multicolumn}1@r{b}^{2}c \\ \mathrm{\backslash cline}4-13\&\&\&\&\&-\&abc\&+\&a{c}^2\&-\&b^{2}c\&+\&\mathrm{\backslash multicolumn}1@r{c}^3 \\ \&\&\&\&\&-\&abc\&\&\&-\&b^{2}c\&-\&\mathrm{\backslash multicolumn}1@rb{c}^2 \\ \mathrm{\backslash cline}6-13\&\&\&\&\&\&\&\&a{c}^2\&+\&b{c}^2\&+\&\mathrm{\backslash multicolumn}1@r{c}^3 \\ \&\&\&\&\&\&\&\&a{c}^2\&+\&b{c}^2\&+\&\mathrm{\backslash multicolumn}1@r{c}^3 \\ \mathrm{\backslash cline}9-13 \end{gathered}$$

Exercise 24.

Divide:

1. $x^2+15x+56$ by $x+7$.

2. $x^2-15x+56$ by $x-7$.

3. $x^2+x-56$ by $x-7$.

4. $x^2-x-56$ by $x+7$.

5. $2a^2+11a+5$ by $2a+1$.

6. $6a^2-7a-3$ by $2a-3$.

7. $4a^2+23a+15$ by $4a+3$.

8. $3a^2-4a-4$ by $2-a$.

9. $x^4+x^2+1$ by $x^2+x+1$.

10. $x^8+x^4+1$ by $x^4-x^2+1$.

11. $1-a^3{b}^3$ by $1-ab$.

12. $x^3-8x-3$ by $x-3$.

13. $a^2-2ab+b^2-c^2$ by $a-b-c$.

14. $a^2+2ab+b^2-c^2$ by $a+b+c$.

15. $x^2-y^2+2yz-z^2$ by $x-y+z$.

16. $c^4+2c^2-c+2$ by $c^2-c+1$.