1. Преобразовать в многочлен:

1) $(x-10)(x+10)$;
2) $(2a+3)(2a-3)$;
3) $(y-5b)(y+5b)$;
4) $(8x+y)(y-8x)$;

5) $(x+7)^2$;
6) $(b+5)^2$;
7) $(a-2x)^2$;
8) $(ab-1)^2$;
9) $(-1-xy)^2$;
10)$(x-y-1)^2$;

2. Разложить на множители:

1) $15ax+20ay$;
2) $36by-9yc$;
3) $x^2-xy$;
4) $xy-y^2$;
5) $a^2+5ab$;
6) $15c-10c^2$;

7) $x^2-25$;
8) $16-c^2$;
9) $a^2-6a+9$;
10) $x^2+8x+16$;
11) $a^3-8$;
12) $b^3+27$;

13) $ab+8a+9b+72$;
14) $6m-12-2n+mn$;
15) $8a^2-50y^2$;
16) $x^3-125$;

3. Указать допустимые значения переменной в выражении:

1) $x^2-8x+9$;
2) $\displaystyle\frac{\displaystyle1}{\displaystyle6x-3}$;
3) $\displaystyle\frac{\displaystyle3x-6}{\displaystyle7}$;
4) $\displaystyle\frac{\displaystyle x^2-8}{\displaystyle4x(x+1)}$;
5) $\displaystyle\frac{\displaystyle x-5}{\displaystyle x^2+25}-3x$;
6) $\displaystyle\frac{\displaystyle x}{\displaystyle x+8}+\displaystyle\frac{\displaystyle x-8}{\displaystyle x}$;

4. Сократить:

1) $\displaystyle\frac{2x}{3x}$;
2) $\displaystyle\frac{15x}{25y}$;
3) $\displaystyle\frac{6a}{24a}$;
4) $\displaystyle\frac{7ab}{21bc}$;
5) $-\displaystyle\frac{2xy}{5x^2y}$;
6) $\displaystyle\frac{8x^2y^2}{24xy}$;
7) $\displaystyle\frac{10xz}{15yz}$;
8) $\displaystyle\frac{6ab^2}{9bc^2}$;
9) $\displaystyle\frac{2ay^3}{-4a^2b}$;

10) $\displaystyle\frac{-6p^2q}{-2q^3}$;
11) $-\displaystyle\frac{ax^2}{xy}$;
12) $\displaystyle\frac{3axy}{6ay^3}$;
13) $\displaystyle\frac{24a^2c^2}{36ac}$;
14) $\displaystyle\frac{63x^2y^3}{42x^6y^4}$;
15) $\displaystyle\frac{8b}{24c}$;
16) $\displaystyle\frac{4a^2}{6ac}$;
17) $\displaystyle\frac{5ay}{15by}$;

18) $\displaystyle\frac{7x^2y}{21xy^2}$;
19) $\displaystyle\frac{a^5b^3}{a^3b^5}$;
20) $\displaystyle\frac{x^6y^4}{x^4y^6}$;
21) $\displaystyle\frac{56m^2n^5}{35mn^5}$;
22) $\displaystyle\frac{25p^4q}{100p^5q}$;
23) $\displaystyle\frac{8^{16}}{8^{12}}$;
24) $\displaystyle\frac{27^{25}}{27^{33}}$;
25) $\displaystyle\frac{3x^4y^5}{3x^4y^5}$;

5. Упростить:

1) $4a^2b^3:(2a^4b^2)$;
2) $3xy^2:(6x^3y^3)$;
3) $36m^2n:(18mn)$;
4) $-32b^5c:(12b^4c^2)$;

5) $-6ax:(-18ax)$;
6) $8a^4b^3:a^3b^2$;
7) $-128x^7y^2z^0:(8x^2y^8)$;
8) $343x^2y^2:96x^8y^2$;

9) $6b^2-(2b+5)(3b-7)$;
10) $16x^2-(4x+0.5)(4x-0.5)$;
11) $2y(y-1.5x)-5(x+4y)(y-x)$;

12) $3(a-2b)(2b+a)-0.5b(a-24b)$;
13) $(a+b)(a-b+1)-(a-b)(a+b-1)$;

14) $(a+3b)(a+b+2)-(a+b)(a+3b+2)$;
15) $(a^2-3a+1)(2a+1)^2$;
16) $(2b+3)(b-2)^3$;

17) $(a-1)^3+3(a-1)^2+3(a-1)+1$;
18) $(a+1)^4+(a-1)^4$;
19) $(b-2)(b^4+2b^3+4b^2+8b+16)$;

20) $(a^2+ab+b^2)(a^2-ab+b^2)(a^4-a^2b^2+b^4)$;
21) $(x+3)^5+(x-3)^5-2x^5$;

6. Упростить:

1) $\displaystyle\frac{a(b-2)}{5(b-2)}$;
2) $\displaystyle\frac{3(x+4)}{c(x+4)}$;
3) $\displaystyle\frac{ab(y+3)}{a^2b(y+3)}$;
4) $\displaystyle\frac{15a(a-b)}{20b(a-b)}$;
5) $\displaystyle\frac{3a+12b}{6ab}$;
6) $\displaystyle\frac{15b-20c}{10b}$;
7) $\displaystyle\frac{2a-4}{3(a-2)}$;
8) $\displaystyle\frac{5x(y+2)}{6y+12}$;

9) $\displaystyle\frac{a-3b}{a^2-3ab}$;
10) $\displaystyle\frac{3x^2+15xy}{x+5y}$;
11) $\displaystyle\frac{y^2-16}{3y+12}$;
12) $\displaystyle\frac{5x-15y}{x^2-9y^2}$;
13) $\displaystyle\frac{(c+2)^2}{7c^2+14c}$;
14) $\displaystyle\frac{6cd-18c}{(d-3)^2}$;
15) $\displaystyle\frac{a^2+10a+25}{a^2-25}$;

16) $\displaystyle\frac{y^2-9}{y^2-6y+9}$;
17) $\displaystyle\frac{a^2-ab+b^2}{a^3+b^3}$;
18) $\displaystyle\frac{a^3-b^3}{a-b}$;
19) $\displaystyle\frac{x(y-z)}{y(y-z)}$;
20) $\displaystyle\frac{x^2-4x+4}{x^2-2x}$;
21) $\displaystyle\frac{3y^2+24y}{y^2+16y+64}$;
22) $\displaystyle\frac{b+2}{b^3+8}$;

23) $\displaystyle\frac{8b^2-8a^2}{a^2-2ab+b^2}$;
24) $\displaystyle\frac{25-a^2}{3a-15}$;
25) $\displaystyle\frac{3-3x}{x^2-2x+1}$;
26) $\displaystyle\frac{(b-2)^3}{(2-b)^2}$;
27) $\displaystyle\frac{ax+bx-ay-by}{bx-by}$;
28) $\displaystyle\frac{ab-3b-2a+6}{15-5a}$;

29) $\displaystyle\frac{a^2-6a+9}{27-a^3}$;
30) $\displaystyle\frac{x^6+x^4}{x^4+x^2}$;
31) $\displaystyle\frac{c^6-c^4}{c^3+c^2}$;
32) $\displaystyle\frac{b^7-b^{10}}{b^5-b^2}$;
33) $\displaystyle\frac{(2a-2b)^2}{a-b}$;
34) $\displaystyle\frac{(3c+9d)^2}{c+3d}$;
35) $\displaystyle\frac{(3x+6y)^2}{5x+10y}$;

36) $\displaystyle\frac{4x^2-y^2}{(10x+5y)^2}$;

7. Найти значение дроби:

1) $\displaystyle\frac{15a^2-10ab}{3ab-2b^2}, a=-2,b=-0.1$;
2) $\displaystyle\frac{9c^2-4d^2}{18c^2d-12cd^2}, c=\displaystyle\frac23,d=\displaystyle\frac12$;

3) $\displaystyle\frac{6x^2+12xy}{5xy+10y^2}, x=\displaystyle\frac23,y=-0.4$;
4) $\displaystyle\frac{x^2+6xy+9y^2}{4x^2+12xy}, x=-0.2,y=-0.6$;

8. Выполнить действие:

1) $\displaystyle\frac{x}{3}+\displaystyle\frac{y}{3}$;
2) $\displaystyle\frac{a}{y}+\displaystyle\frac{2a}{y}$;
3) $\displaystyle\frac{a}{5}-\displaystyle\frac{b}{5}$;
4) $\displaystyle\frac{5b^2}{a}-\displaystyle\frac{13b^2}{a}$;
5) $\displaystyle\frac{x+y}{9}-\displaystyle\frac{x}{9}$;
6) $\displaystyle\frac{2c-x}{b}-\displaystyle\frac{x}{b}$;
7) $\displaystyle\frac{m}{p}-\displaystyle\frac{m-p}{p}$;

8) $\displaystyle\frac{11x-5}{14x}+\displaystyle\frac{3x-2}{14x}$; 9)
$\displaystyle\frac{2x-3y}{4xy}+\displaystyle\frac{11y-2x}{4xy}$; 10)
$\displaystyle\frac{7y-5}{12y}-\displaystyle\frac{10y-19}{12y}+\displaystyle\frac{10-15y}{12y}$; 11)
$\displaystyle\frac{11a-2b}{4a}+\displaystyle\frac{2a-3b}{4a}-\displaystyle\frac{a-b}{4a}$;

12) $\displaystyle\frac{3x-y^4}{4y^5}-\displaystyle\frac{y^4+3x}{4y^5}$;
13) $\displaystyle\frac{12p-1}{3p^2}-\displaystyle\frac{1-3p}{3p^2}$;
14) $\displaystyle\frac{5c-2d}{4c}-\displaystyle\frac{3d}{4c}+\displaystyle\frac{d-5c}{4c}$;
15) $\displaystyle\frac{2a}{b}-\displaystyle\frac{1-6a}{b}+\displaystyle\frac{13-8a}b$;

16) $\displaystyle\frac{4b-2}{3b}-\displaystyle\frac{2b-1}{3b}+\displaystyle\frac1{3b}$;
17) $\displaystyle\frac{x}{y-1}+\displaystyle\frac{5}{1-y}$;
18) $\displaystyle\frac{a}{c-3}-\displaystyle\frac{6}{3-c}$;
19) $\displaystyle\frac{2m}{m-n}+\displaystyle\frac{2n}{n-m}$;
20) $\displaystyle\frac{a^2+16}{a-4}+\displaystyle\frac{8a}{4-a}$;

21) $\displaystyle\frac{x^2+9y^2}{x-3y}+\displaystyle\frac{6xy}{3y-x}$;

9. Пользуясь равенством $\displaystyle\frac{a+b}c=\displaystyle\frac{a}c+\displaystyle\frac{b}c$, представить в виде суммы или разности дробей:

1) $\displaystyle\frac{a+b}{x}$;
2) $\displaystyle\frac{2a^2+a}{y}$;
3) $\displaystyle\frac{x^2+6y^2}{2xy}$;
4) $\displaystyle\frac{12a+y^2}{6ay}$;
5) $\displaystyle\frac{x^2+y^2}{x^4}$;
6) $\displaystyle\frac{2x-y}{b}$;
7) $\displaystyle\frac{a^2+1}{2a}$;
8) $\displaystyle\frac{a^2-3ab}{a^3}$;