## 2020年1月21日火曜日

### 数学 - 代数学 - 因数分解と分数式 - 整式の加法、乗法、除法、通分、最小公倍数、因数分解、既約

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

2. $\begin{array}{l}\frac{1}{x\left(x+1\right)}+\frac{1}{\left(x+1\right)\left(x+2\right)}+\frac{1}{\left(x+2\right)\left(x+3\right)}\\ =\frac{\left(x+2\right)\left(x+3\right)+x\left(x+3\right)+x\left(x+1\right)}{x\left(x+1\right)\left(x+2\right)\left(x+3\right)}\\ =\frac{{x}^{2}+5x+6+{x}^{2}+3x+{x}^{2}+x}{x\left(x+1\right)\left(x+2\right)\left(x+3\right)}\\ =\frac{3{x}^{2}+2{x}^{2}+6x+6}{x\left(x+1\right)\left(x+2\right)\left(x+3\right)}\end{array}$

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

4. $\begin{array}{l}\frac{1-\left(a+4\right)+a+3}{\left(a+2\right)\left(a+3\right)\left(a+4\right)}\\ =0\end{array}$

5. $\begin{array}{l}\frac{\left(x+y-z\right)\left(x-y+z\right)}{\left(x+y+z\right)\left(x+y-z\right)}+\frac{\left(y+z-x\right)\left(y-z+x\right)}{\left(y+z+x\right)\left(y+z-x\right)}\\ +\frac{\left(z+x-y\right)\left(z-x+y\right)}{\left(z+x+y\right)\left(z+x-y\right)}\\ =\frac{x-y+z}{x+y+z}+\frac{y-z+x}{y+z+x}+\frac{z-x+y}{z+x+y}\\ =\frac{x+y+z}{x+y+z}\\ =1\end{array}$

6. $\begin{array}{l}1+\frac{1}{x+1}-1-\frac{1}{x+2}+1+\frac{1}{x+3}-1-\frac{1}{x+4}\\ =\frac{1}{x+1}-\frac{1}{x+2}+\frac{1}{x+3}-\frac{1}{x+4}\\ =\frac{1}{\left(x+1\right)\left(x+2\right)}+\frac{1}{x+3}-\frac{1}{x+4}\\ =\frac{x+3+{x}^{2}+3x+2}{\left(x+1\right)\left(x+2\right)\left(x+3\right)}-\frac{1}{x+4}\\ =\frac{{x}^{2}+4x+5}{\left(x+1\right)\left(x+2\right)\left(x+3\right)}-\frac{1}{x+4}\\ =\frac{{x}^{3}+8{x}^{2}+21x+20-\left({x}^{2}+3x+2\right)\left(x+3\right)}{\left(x+1\right)\left(x+2\right)\left(x+3\right)\left(x+4\right)}\\ =\frac{{x}^{3}+8{x}^{2}+21x+20-{x}^{3}-6{x}^{2}-12x-6}{\left(x+1\right)\left(x+2\right)\left(x+3\right)\left(x+4\right)}\\ =\frac{2{x}^{2}+9x+14}{\left(x+1\right)\left(x+2\right)\left(x+3\right)\left(x+4\right)}\end{array}$

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

8. $\begin{array}{l}\frac{\left(2x-1\right)\left(x-2\right)}{\left(x+6\right)\left(x-5\right)}·\frac{\left(x+2\right)\left(x+6\right)}{\left(2x-1\right)\left(x+5\right)}\\ =\frac{\left(x-2\right)\left(x+2\right)}{\left(x-5\right)\left(x+5\right)}\\ =\frac{{x}^{2}-4}{{x}^{2}-25}\end{array}$

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

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

11. $\begin{array}{l}\frac{{x}^{2}-4x+3-4x+12+12x-12}{\left(x-1\right)\left(x-3\right)}·\frac{{x}^{2}+4x+3+4x+12-12x-12}{\left(x+1\right)\left(x+3\right)}\\ =\frac{{x}^{2}+4x+3}{\left(x-1\right)\left(x-3\right)}·\frac{{x}^{2}-4x+3}{\left(x+1\right)\left(x+3\right)}\\ =1\end{array}$

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

13. $\begin{array}{l}\frac{-\left(b-c\right)\left(b+1\right)\left(c+1\right)-\left(c-a\right)\left(c+1\right)\left(a+1\right)-\left(a-b\right)\left(a+1\right)\left(b+1\right)}{\left(a-b\right)\left(b-c\right)\left(c-a\right)\left(a+1\right)\left(b+1\right)\left(c+1\right)}\\ =\frac{\left(c+1-b-1\right){a}^{2}+\left(-c\left(c+1\right)+c+1-\left(b+1\right)+b\left(b+1\right)\right)a}{\left(a-b\right)\left(b-c\right)\left(c-a\right)\left(a+1\right)\left(b+1\right)\left(c+1\right)}\\ +\frac{-\left(b-c\right)\left(b+1\right)\left(c+1\right)-c\left(c+1\right)+b\left(b+1\right)}{\left(a-b\right)\left(b-c\right)\left(c-a\right)\left(a+1\right)\left(b+1\right)\left(c+1\right)}\\ =\frac{\left(c-b\right){a}^{2}+\left({b}^{2}-{c}^{2}\right)a+\left(1-c-1\right){b}^{2}+\left(1-c-1+c\left(c+1\right)\right)b+c\left(c+1\right)-c\left(c+1\right)}{\left(a-b\right)\left(b-c\right)\left(c-a\right)\left(a+1\right)\left(b+1\right)\left(c+1\right)}\\ =\frac{-\left(b-c\right){a}^{2}+\left(b-c\right)\left(b+c\right)a-c{b}^{2}+{c}^{2}b}{\left(a-b\right)\left(b-c\right)\left(c-a\right)\left(a+1\right)\left(b+1\right)\left(c+1\right)}\\ =\frac{-\left(b-c\right){a}^{2}+\left(b-c\right)\left(b+c\right)a-bc\left(b-c\right)}{\left(\alpha -b\right)\left(b-c\right)\left(c-a\right)\left(a+1\right)\left(b+1\right)\left(c+1\right)}\\ =\frac{-{a}^{2}+\left(b+c\right)a-bc}{\left(a-b\right)\left(c-a\right)\left(a+1\right)\left(b+1\right)\left(c+1\right)}\\ =\frac{-\left(a-b\right)\left(a-c\right)}{\left(a-b\right)\left(c-a\right)\left(a+1\right)\left(b+1\right)\left(c+1\right)}\\ =\frac{1}{\left(a+1\right)\left(b+1\right)\left(c+1\right)}\end{array}$

14. $\begin{array}{l}1-\frac{1}{1-\frac{1}{1-\frac{1}{\frac{x-1}{x}}}}\\ =1-\frac{1}{1-\frac{1}{1-\frac{x}{x-1}}}\\ =1-\frac{1}{1-\frac{1}{\frac{-1}{x-1}}}\\ =1-\frac{1}{1+x-1}\\ =1-\frac{1}{x}\\ =\frac{x-1}{x}\end{array}$

15. $\begin{array}{l}\frac{\frac{x+6}{x+2}}{{x}^{2}-10-\frac{{x}^{3}+8}{\frac{{x}^{2}+4x+4}{x+4}}}\\ =\frac{x+6}{x+2}·\frac{1}{{x}^{2}-10-\frac{\left(x+2\right)\left({x}^{2}-2x+4\right)\left(x+4\right)}{{\left(x+2\right)}^{2}}}\\ =\frac{x+6}{x+2}·\frac{1}{{x}^{2}-10-\frac{\left({x}^{2}-2x+4\right)\left(x+4\right)}{x+2}}\\ =\frac{x+6}{\left({x}^{2}-10\right)\left(x+2\right)-\left({x}^{3}+2{x}^{2}-4x+16\right)}\\ =\frac{x+6}{-6x-36}\\ =\frac{x+6}{-6\left(x+6\right)}\\ =-\frac{1}{6}\end{array}$