## 2020年1月19日日曜日

### 数学 - 代数学 - 因数分解と分数式 - いろいろな整式の因数分解の方法

1. $\begin{array}{l}{x}^{3}-{x}^{2}y-x+y\\ ={x}^{2}\left(x-y\right)-\left(x-y\right)\\ =\left({x}^{2}-1\right)\left(x-y\right)\\ =\left(x-y\right)\left(x+1\right)\left(x-1\right)\end{array}$

2. $\begin{array}{l}{a}^{3}-{a}^{2}b-a{b}^{2}+{b}^{3}\\ ={\left(a+b\right)}^{3}-4{a}^{2}b-4a{b}^{2}\\ ={\left(a+b\right)}^{3}-4ab\left(a+b\right)\\ =\left(a+b\right)\left({\left(a+b\right)}^{2}-4ab\right)\\ =\left(a+b\right)\left({a}^{2}-2ab+{b}^{2}\right)\\ =\left(a+b\right){\left(a-b\right)}^{2}\end{array}$

3. $\begin{array}{l}{a}^{2}-2ab+{b}^{2}-9\\ ={\left(a-b\right)}^{2}-{3}^{2}\\ =\left(a-b+3\right)\left(a-b-3\right)\end{array}$

4. $\begin{array}{l}{a}^{2}-9{b}^{2}+4{c}^{2}-4ac\\ ={\left(a-2c\right)}^{2}-{\left(3b\right)}^{2}\\ =\left(a+3b-2c\right)\left(a-2b-2c\right)\end{array}$

5. $\begin{array}{l}{x}^{2}y+{y}^{2}z-{y}^{3}-{x}^{2}z\\ =-\left({x}^{2}-{y}^{2}\right)z+\left({x}^{2}-{y}^{2}\right)y\\ =\left({x}^{2}-{y}^{2}\right)\left(y-z\right)\\ =\left(x+y\right)\left(x-y\right)\left(y-z\right)\end{array}$

6. $\begin{array}{l}\left(x+y\right)\left(x+y-3\right)-4\\ ={\left(x+y\right)}^{2}-3\left(x+y\right)-4\\ =\left(x+y-4\right)\left(x+y+1\right)\end{array}$

7. $\begin{array}{l}2{\left(a-2b\right)}^{2}-7\left(a-2b\right)+6\\ =\left(a-2b-2\right)\left(2a-4b-3\right)\end{array}$

8. $\begin{array}{l}{x}^{2}-\left(5a-4b\right)x-20ab\\ =\left(x+4a\right)\left(x-5a\right)\end{array}$

9. $\begin{array}{l}{x}^{2}+xy-6{y}^{2}+5x+35y-36\\ ={x}^{2}+\left(5+y\right)x-6{y}^{2}+35y-36\\ ={x}^{2}+\left(y+5\right)x-\left(2y-9\right)\left(3y-4\right)\\ =\left(x-2y+9\right)\left(x+3y-4\right)\end{array}$

10. $\begin{array}{l}10{x}^{2}-48xy-10{y}^{2}+31x+y+3\\ =10{x}^{2}+\left(31-48y\right)x-\left(10{y}^{2}-y-3\right)\\ =10{x}^{2}-\left(48y-31\right)x-\left(2y-1\right)\left(5y+3\right)\\ =\left(10x+2y-1\right)\left(x-5y-3\right)\end{array}$

11. $\begin{array}{l}{x}^{3}-{x}^{2}y-30x{y}^{2}\\ =x\left({x}^{2}-xy-30{y}^{2}\right)\\ =x\left(x+5y\right)\left(x-6y\right)\end{array}$

12. $\begin{array}{l}{\left(2x-y\right)}^{3}-{\left(2y-x\right)}^{3}\\ =\left(\left(2x-y\right)-\left(2y-x\right)\right)\left({\left(2x-y\right)}^{2}+\left(2x-y\right)\left(2y-x\right)+{\left(2y-x\right)}^{2}\right)\\ =\left(3x-3y\right)\left(4{x}^{2}-4xy+{y}^{2}+4xy-2{x}^{2}-2{y}^{2}+xy+4{y}^{2}-4xy+{x}^{2}\right)\\ =3\left(x-y\right)\left(3{x}^{2}-3xy+3{y}^{2}\right)\\ =9\left(x-y\right)\left({x}^{2}-xy+{y}^{2}\right)\end{array}$

13. $\begin{array}{l}{\left({x}^{2}+4x\right)}^{2}-8\left({x}^{2}+4x\right)-48\\ =\left({x}^{2}+4x-12\right)\left({x}^{2}+4x+4\right)\\ =\left(x+6\right)\left(x-2\right){\left(x+2\right)}^{2}\end{array}$

14. $\begin{array}{l}\left({a}^{2}-{b}^{2}\right){x}^{2}+4abx-\left({a}^{2}-{b}^{2}\right)\\ =\left(a+b\right)\left(a-b\right){x}^{2}+4abx-\left(a+b\right)\left(a-b\right)\\ =\left(\left(a+b\right)x-\left(a-b\right)\right)\left(\left(a-b\right)x+\left(a+b\right)\right)\\ =\left(ax+bx-a+b\right)\left(ax-bx+a+b\right)\end{array}$

15. $\begin{array}{l}{x}^{4}-2{x}^{2}+1\\ ={\left({x}^{2}-1\right)}^{2}\\ =\left(x+1{\right)}^{2}{\left(x-1\right)}^{2}\end{array}$

16. $\begin{array}{l}{x}^{4}-13{x}^{2}+36\\ =\left({x}^{2}-4\right)\left({x}^{2}-9\right)\\ =\left(x+2\right)\left(x-2\right)\left(x+3\right)\left(x-3\right)\end{array}$

17. $\begin{array}{l}{x}^{4}-26{x}^{2}{y}^{2}+25{y}^{4}\\ =\left({x}^{2}-25{y}^{2}\right)\left({x}^{2}-{y}^{2}\right)\\ =\left(x+5y\right)\left(x-5y\right)\left(x+y\right)\left(x-y\right)\end{array}$

18. $\begin{array}{l}{a}^{4}+\frac{1}{4}{a}^{2}+\frac{1}{16}\\ =\frac{1}{16}\left(16{a}^{4}+4{a}^{2}+1\right)\\ =\frac{1}{16}\left({\left(4{a}^{2}+1\right)}^{2}-4{a}^{2}\right)\\ =\frac{1}{16}\left(4{a}^{2}+2a+1\right)\left(4{a}^{2}-2a+1\right)\end{array}$

19. $\begin{array}{l}9{x}^{4}-34{x}^{2}{y}^{2}+25{y}^{4}\\ =\left({x}^{2}-{y}^{2}\right)\left(9{x}^{2}-25{y}^{2}\right)\\ =\left(x+y\right)\left(x-y\right)\left(3x+5y\right)\left(3x-5y\right)\end{array}$

20. $\begin{array}{l}{\left(ax-by\right)}^{2}+{\left(ay+bx\right)}^{2}\\ ={a}^{2}{x}^{2}-2abxy+{b}^{2}{y}^{2}+{a}^{2}{y}^{2}+2abxy+{b}^{2}{x}^{2}\\ ={a}^{2}{x}^{2}+{b}^{2}{y}^{2}+{a}^{2}{y}^{2}+{b}^{2}{x}^{2}\\ =\left({a}^{2}+{b}^{2}\right){x}^{2}+\left({a}^{2}+{b}^{2}\right){y}^{2}\\ =\left({a}^{2}+{b}^{2}\right)\left({x}^{2}+{y}^{2}\right)\end{array}$

21. $\begin{array}{l}{x}^{8}-{y}^{8}\\ =\left({x}^{4}+{y}^{4}\right)\left({x}^{4}-{y}^{4}\right)\\ =\left({x}^{4}+{y}^{4}\right)\left({x}^{2}+{y}^{2}\right)\left({x}^{2}-{y}^{2}\right)\\ =\left({x}^{4}+{y}^{4}\right)\left({x}^{2}+{y}^{2}\right)\left(x+y\right)\left(x-y\right)\end{array}$

22. $\begin{array}{l}{a}^{2}{b}^{2}-{a}^{2}-{b}^{2}-4ab+1\\ =\left({a}^{2}-1\right){b}^{2}-4ab-\left({a}^{2}-1\right)\\ =\left(a+1\right)\left(a-1\right){b}^{2}-4ab-\left(a+1\right)\left(a-1\right)\\ =\left(\left(a+1\right)b+\left(a-1\right)\right)\left(\left(a-1\right)b-\left(a+1\right)\right)\\ =\left(ab+a+b-1\right)\left(ab-a-b-1\right)\end{array}$

23. $\begin{array}{l}{a}^{2}bc+abd+bc-a{b}^{2}-a{c}^{2}-cd\\ =\left(ab-c\right)d+\left({a}^{2}bc+bc-a{b}^{2}-a{c}^{2}\right)\\ =\left(ab-c\right)d-\left(a{b}^{2}+a{c}^{2}-bc-{a}^{2}bc\right)\\ =\left(ab-c\right)d-\left(a{b}^{2}-\left(c+{a}^{2}c\right)b+a{c}^{2}\right)\\ =\left(ab-c\right)d-\left(ab-c\right)\left(b-ac\right)\\ =\left(ab-c\right)\left(d-b+ac\right)\\ =\left(ab-c\right)\left(ac-b+d\right)\end{array}$

24. $\begin{array}{l}{x}^{2}-{y}^{2}+2yz+2zx+4x+2y+2z+3\\ =\left(2x+2y+2\right)z+{x}^{2}-{y}^{2}+4x+2y+3\\ =2\left(x+y+1\right)z+{x}^{2}+4x-\left({y}^{2}-2y-3\right)\\ =2\left(x+y+1\right)z+{x}^{2}+4x-\left(y-3\right)\left(y+1\right)\\ =2\left(x+y+1\right)+\left(x-y+3\right)\left(x+y+1\right)\\ =\left(x+y+1\right)\left(x-y+5\right)\end{array}$

25. $\begin{array}{l}4-4{a}^{3}-{b}^{2}+{a}^{3}{b}^{2}\\ =4\left(1-{a}^{3}\right)-{b}^{2}\left(1-{a}^{3}\right)\\ =\left(1-{a}^{3}\right)\left(4-{b}^{2}\right)\\ =\left({a}^{3}-1\right)\left({b}^{2}-4\right)\\ =\left(a-1\right)\left({a}^{2}+a+1\right)\left(b+2\right)\left(b-2\right)\end{array}$

26. $\begin{array}{l}{\left(a-b\right)}^{3}+{\left(b-c\right)}^{3}+{\left(c-a\right)}^{3}\\ =\left(a-b+b-c+c-a\right)R\left(a,b,c\right)+3\left(a-b\right)\left(b-c\right)\left(c-a\right)\\ =3\left(a-b\right)\left(b-c\right)\left(c-a\right)\end{array}$

27. $\begin{array}{l}{a}^{3}\left(b-c\right)+{b}^{3}\left(c-a\right)+{c}^{3}\left(a-b\right)\\ =\left(b-c\right){a}^{3}-\left({b}^{3}-{c}^{3}\right)a+{b}^{3}c-b{c}^{3}\\ =\left(b-c\right){a}^{3}-\left(b-c\right)\left({b}^{2}+bc+{c}^{2}\right)a+bc\left({b}^{2}-{c}^{2}\right)\\ =\left(b-c\right)\left({a}^{3}-\left({b}^{2}+bc+{c}^{2}\right)a+bc\left(b+c\right)\right)\\ =\left(b-c\right)\left({a}^{3}-a{b}^{2}-abc-a{c}^{2}+{b}^{2}c+b{c}^{2}\right)\\ =\left(b-c\right)\left(\left(c-a\right){b}^{2}+\left({c}^{2}-ac\right)b+{a}^{3}-a{c}^{2}\right)\\ =\left(b-c\right)\left(\left(c-a\right){b}^{2}+c\left(c-a\right)b+a\left({a}^{2}-{c}^{2}\right)\right)\\ =\left(b-c\right)\left(c-a\right)\left({b}^{2}+bc-a\left(c+a\right)\right)\\ =\left(b-c\right)\left(c-a\right)\left(\left(b-a\right)c+{b}^{2}-{a}^{2}\right)\\ =\left(b-c\right)\left(c-a\right)\left(b-a\right)\left(c+b+a\right)\\ =-\left(a+b+c\right)\left(a-b\right)\left(b-c\right)\left(c-a\right)\end{array}$

28. $\begin{array}{l}bc\left(b+c\right)+ca\left(c+a\right)+ab\left(a+b\right)+2abc\\ =\left(c+b\right){a}^{2}+\left({c}^{2}+{b}^{2}+2bc\right)a+bc\left(b+c\right)\\ =\left(b+c\right)\left({a}^{2}+\left(b+c\right)a+bc\right)\\ =\left(b+c\right)\left(a+b\right)\left(a+c\right)\\ =\left(a+b\right)\left(b+c\right)\left(c+a\right)\end{array}$

29. $\begin{array}{l}4{\left(ad+bc\right)}^{2}-{\left({a}^{2}-{b}^{2}-{c}^{2}+{d}^{2}\right)}^{2}\\ =\left(2\left(ad+bc\right)-\left({a}^{2}-{b}^{2}-{c}^{2}+{d}^{2}\right)\right)\left(2\left(ad+bc\right)+\left({a}^{2}-{b}^{2}-{c}^{2}+{d}^{2}\right)\right)\\ =\left({\left(b+c\right)}^{2}-{\left(a-d\right)}^{2}\right)\left({\left(a+d\right)}^{2}-{\left(b-c\right)}^{2}\right)\\ =\left(b+c+a-d\right)\left(b+c-a+d\right)\left(a+d+b-c\right)\left(a+d-b+c\right)\\ =\left(b+c+d-a\right)\left(a+c+d-b\right)\left(a+b+d-c\right)\left(a+b+c-d\right)\end{array}$

30. $\begin{array}{l}2\left({b}^{2}{c}^{2}+{c}^{2}{a}^{2}+{a}^{2}{b}^{2}\right)-\left({a}^{4}+{b}^{4}+{c}^{4}\right)\\ =-{a}^{4}+2\left({b}^{2}+{c}^{2}\right){a}^{2}+2{b}^{2}{c}^{2}-{b}^{4}-{c}^{4}\\ =-\left({a}^{4}-2\left({b}^{2}+{c}^{2}\right){a}^{2}+{b}^{4}-2{b}^{2}{c}^{2}+{c}^{4}\right)\\ =-\left({a}^{4}-2\left({b}^{2}+{c}^{2}\right){a}^{2}+{\left({b}^{2}+{c}^{2}\right)}^{2}-4{b}^{2}{c}^{2}\right)\\ =-\left({a}^{4}-2\left({b}^{2}+{c}^{2}\right){a}^{2}+\left({b}^{2}+{c}^{2}+2bc\right)\left({b}^{2}+{c}^{2}-2bc\right)\right)\\ =-\left({a}^{2}-\left({b}^{2}+{c}^{2}+2bc\right)\right)\left({a}^{2}-\left({b}^{2}+{c}^{2}-2bc\right)\right)\\ =-\left({a}^{2}-{\left(b+c\right)}^{2}\right)\left({a}^{2}-{\left(b-c\right)}^{2}\right)\\ =-\left(a-b-c\right)\left(a+b+c\right)\left(a-b+c\right)\left(a+b-c\right)\\ =\left(a+b+c\right)\left(-a+b+c\right)\left(a-b+c\right)\left(a+b-c\right)\end{array}$