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

代数への出発
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代数への出発 (新装版数学入門シリーズ) (松坂和夫(著)、岩波書店)の第3章(因数分解と分数式)、練習問題の問1の解答を求めてみる。

1. $\begin{array}{l} x^{3} - x^{2} y - x + y \\ = x^{2} (x - y) - (x - y) \\ = (x^{2} - 1) (x - y) \\ = (x - y) (x + 1) (x - 1) \end{array}$
2. $\begin{array}{l} a^{3} - a^{2} b - a b^{2} + b^{3} \\ = {(a + b)}^{3} - 4 a^{2} b - 4 a b^{2} \\ = {(a + b)}^{3} - 4 a b (a + b) \\ = (a + b) ({(a + b)}^{2} - 4 a b) \\ = (a + b) (a^{2} - 2 a b + b^{2}) \\ = (a + b) {(a - b)}^{2} \end{array}$
3. $\begin{array}{l} a^{2} - 2 a b + b^{2} - 9 \\ = {(a - b)}^{2} - 3^{2} \\ = (a - b + 3) (a - b - 3) \end{array}$
4. $\begin{array}{l} a^{2} - 9 b^{2} + 4 c^{2} - 4 a c \\ = {(a - 2 c)}^{2} - {(3 b)}^{2} \\ = (a + 3 b - 2 c) (a - 2 b - 2 c) \end{array}$
5. $\begin{array}{l} x^{2} y + y^{2} z - y^{3} - x^{2} z \\ = - (x^{2} - y^{2}) z + (x^{2} - y^{2}) y \\ = (x^{2} - y^{2}) (y - z) \\ = (x + y) (x - y) (y - z) \end{array}$
6. $\begin{array}{l} (x + y) (x + y - 3) - 4 \\ = {(x + y)}^{2} - 3 (x + y) - 4 \\ = (x + y - 4) (x + y + 1) \end{array}$
7. $\begin{array}{l} 2 {(a - 2 b)}^{2} - 7 (a - 2 b) + 6 \\ = (a - 2 b - 2) (2 a - 4 b - 3) \end{array}$
8. $\begin{array}{l} x^{2} - (5 a - 4 b) x - 20 a b \\ = (x + 4 a) (x - 5 a) \end{array}$
9. $\begin{array}{l} x^{2} + x y - 6 y^{2} + 5 x + 35 y - 36 \\ = x^{2} + (5 + y) x - 6 y^{2} + 35 y - 36 \\ = x^{2} + (y + 5) x - (2 y - 9) (3 y - 4) \\ = (x - 2 y + 9) (x + 3 y - 4) \end{array}$
10. $\begin{array}{l} 10 x^{2} - 48 x y - 10 y^{2} + 31 x + y + 3 \\ = 10 x^{2} + (31 - 48 y) x - (10 y^{2} - y - 3) \\ = 10 x^{2} - (48 y - 31) x - (2 y - 1) (5 y + 3) \\ = (10 x + 2 y - 1) (x - 5 y - 3) \end{array}$
11. $\begin{array}{l} x^{3} - x^{2} y - 30 x y^{2} \\ = x (x^{2} - x y - 30 y^{2}) \\ = x (x + 5 y) (x - 6 y) \end{array}$
12. $\begin{array}{l} {(2 x - y)}^{3} - {(2 y - x)}^{3} \\ = ((2 x - y) - (2 y - x)) ({(2 x - y)}^{2} + (2 x - y) (2 y - x) + {(2 y - x)}^{2}) \\ = (3 x - 3 y) (4 x^{2} - 4 x y + y^{2} + 4 x y - 2 x^{2} - 2 y^{2} + x y + 4 y^{2} - 4 x y + x^{2}) \\ = 3 (x - y) (3 x^{2} - 3 x y + 3 y^{2}) \\ = 9 (x - y) (x^{2} - x y + y^{2}) \end{array}$
13. $\begin{array}{l} {(x^{2} + 4 x)}^{2} - 8 (x^{2} + 4 x) - 48 \\ = (x^{2} + 4 x - 12) (x^{2} + 4 x + 4) \\ = (x + 6) (x - 2) {(x + 2)}^{2} \end{array}$
14. $\begin{array}{l} (a^{2} - b^{2}) x^{2} + 4 a b x - (a^{2} - b^{2}) \\ = (a + b) (a - b) x^{2} + 4 a b x - (a + b) (a - b) \\ = ((a + b) x - (a - b)) ((a - b) x + (a + b)) \\ = (a x + b x - a + b) (a x - b x + a + b) \end{array}$
15. $\begin{array}{l} x^{4} - 2 x^{2} + 1 \\ = {(x^{2} - 1)}^{2} \\ = (x + 1)^{2} {(x - 1)}^{2} \end{array}$
16. $\begin{array}{l} x^{4} - 13 x^{2} + 36 \\ = (x^{2} - 4) (x^{2} - 9) \\ = (x + 2) (x - 2) (x + 3) (x - 3) \end{array}$
17. $\begin{array}{l} x^{4} - 26 x^{2} y^{2} + 25 y^{4} \\ = (x^{2} - 25 y^{2}) (x^{2} - y^{2}) \\ = (x + 5 y) (x - 5 y) (x + y) (x - y) \end{array}$
18. $\begin{array}{l} a^{4} + \frac{1}{4} a^{2} + \frac{1}{16} \\ = \frac{1}{16} (16 a^{4} + 4 a^{2} + 1) \\ = \frac{1}{16} ({(4 a^{2} + 1)}^{2} - 4 a^{2}) \\ = \frac{1}{16} (4 a^{2} + 2 a + 1) (4 a^{2} - 2 a + 1) \end{array}$
19. $\begin{array}{l} 9 x^{4} - 34 x^{2} y^{2} + 25 y^{4} \\ = (x^{2} - y^{2}) (9 x^{2} - 25 y^{2}) \\ = (x + y) (x - y) (3 x + 5 y) (3 x - 5 y) \end{array}$
20. $\begin{array}{l} {(a x - b y)}^{2} + {(a y + b x)}^{2} \\ = a^{2} x^{2} - 2 a b x y + b^{2} y^{2} + a^{2} y^{2} + 2 a b x y + b^{2} x^{2} \\ = a^{2} x^{2} + b^{2} y^{2} + a^{2} y^{2} + b^{2} x^{2} \\ = (a^{2} + b^{2}) x^{2} + (a^{2} + b^{2}) y^{2} \\ = (a^{2} + b^{2}) (x^{2} + y^{2}) \end{array}$
21. $\begin{array}{l} x^{8} - y^{8} \\ = (x^{4} + y^{4}) (x^{4} - y^{4}) \\ = (x^{4} + y^{4}) (x^{2} + y^{2}) (x^{2} - y^{2}) \\ = (x^{4} + y^{4}) (x^{2} + y^{2}) (x + y) (x - y) \end{array}$
22. $\begin{array}{l} a^{2} b^{2} - a^{2} - b^{2} - 4 a b + 1 \\ = (a^{2} - 1) b^{2} - 4 a b - (a^{2} - 1) \\ = (a + 1) (a - 1) b^{2} - 4 a b - (a + 1) (a - 1) \\ = ((a + 1) b + (a - 1)) ((a - 1) b - (a + 1)) \\ = (a b + a + b - 1) (a b - a - b - 1) \end{array}$
23. $\begin{array}{l} a^{2} b c + a b d + b c - a b^{2} - a c^{2} - c d \\ = (a b - c) d + (a^{2} b c + b c - a b^{2} - a c^{2}) \\ = (a b - c) d - (a b^{2} + a c^{2} - b c - a^{2} b c) \\ = (a b - c) d - (a b^{2} - (c + a^{2} c) b + a c^{2}) \\ = (a b - c) d - (a b - c) (b - a c) \\ = (a b - c) (d - b + a c) \\ = (a b - c) (a c - b + d) \end{array}$
24. $\begin{array}{l} x^{2} - y^{2} + 2 y z + 2 z x + 4 x + 2 y + 2 z + 3 \\ = (2 x + 2 y + 2) z + x^{2} - y^{2} + 4 x + 2 y + 3 \\ = 2 (x + y + 1) z + x^{2} + 4 x - (y^{2} - 2 y - 3) \\ = 2 (x + y + 1) z + x^{2} + 4 x - (y - 3) (y + 1) \\ = 2 (x + y + 1) + (x - y + 3) (x + y + 1) \\ = (x + y + 1) (x - y + 5) \end{array}$
25. $\begin{array}{l} 4 - 4 a^{3} - b^{2} + a^{3} b^{2} \\ = 4 (1 - a^{3}) - b^{2} (1 - a^{3}) \\ = (1 - a^{3}) (4 - b^{2}) \\ = (a^{3} - 1) (b^{2} - 4) \\ = (a - 1) (a^{2} + a + 1) (b + 2) (b - 2) \end{array}$
26. $\begin{array}{l} {(a - b)}^{3} + {(b - c)}^{3} + {(c - a)}^{3} \\ = (a - b + b - c + c - a) R (a, b, c) + 3 (a - b) (b - c) (c - a) \\ = 3 (a - b) (b - c) (c - a) \end{array}$
27. $\begin{array}{l} a^{3} (b - c) + b^{3} (c - a) + c^{3} (a - b) \\ = (b - c) a^{3} - (b^{3} - c^{3}) a + b^{3} c - b c^{3} \\ = (b - c) a^{3} - (b - c) (b^{2} + b c + c^{2}) a + b c (b^{2} - c^{2}) \\ = (b - c) (a^{3} - (b^{2} + b c + c^{2}) a + b c (b + c)) \\ = (b - c) (a^{3} - a b^{2} - a b c - a c^{2} + b^{2} c + b c^{2}) \\ = (b - c) ((c - a) b^{2} + (c^{2} - a c) b + a^{3} - a c^{2}) \\ = (b - c) ((c - a) b^{2} + c (c - a) b + a (a^{2} - c^{2})) \\ = (b - c) (c - a) (b^{2} + b c - a (c + a)) \\ = (b - c) (c - a) ((b - a) c + b^{2} - a^{2}) \\ = (b - c) (c - a) (b - a) (c + b + a) \\ = - (a + b + c) (a - b) (b - c) (c - a) \end{array}$
28. $\begin{array}{l} b c (b + c) + c a (c + a) + a b (a + b) + 2 a b c \\ = (c + b) a^{2} + (c^{2} + b^{2} + 2 b c) a + b c (b + c) \\ = (b + c) (a^{2} + (b + c) a + b c) \\ = (b + c) (a + b) (a + c) \\ = (a + b) (b + c) (c + a) \end{array}$
29. $\begin{array}{l} 4 {(a d + b c)}^{2} - {(a^{2} - b^{2} - c^{2} + d^{2})}^{2} \\ = (2 (a d + b c) - (a^{2} - b^{2} - c^{2} + d^{2})) (2 (a d + b c) + (a^{2} - b^{2} - c^{2} + d^{2})) \\ = ({(b + c)}^{2} - {(a - d)}^{2}) ({(a + d)}^{2} - {(b - c)}^{2}) \\ = (b + c + a - d) (b + c - a + d) (a + d + b - c) (a + d - b + c) \\ = (b + c + d - a) (a + c + d - b) (a + b + d - c) (a + b + c - d) \end{array}$
30. $\begin{array}{l} 2 (b^{2} c^{2} + c^{2} a^{2} + a^{2} b^{2}) - (a^{4} + b^{4} + c^{4}) \\ = - a^{4} + 2 (b^{2} + c^{2}) a^{2} + 2 b^{2} c^{2} - b^{4} - c^{4} \\ = - (a^{4} - 2 (b^{2} + c^{2}) a^{2} + b^{4} - 2 b^{2} c^{2} + c^{4}) \\ = - (a^{4} - 2 (b^{2} + c^{2}) a^{2} + {(b^{2} + c^{2})}^{2} - 4 b^{2} c^{2}) \\ = - (a^{4} - 2 (b^{2} + c^{2}) a^{2} + (b^{2} + c^{2} + 2 b c) (b^{2} + c^{2} - 2 b c)) \\ = - (a^{2} - (b^{2} + c^{2} + 2 b c)) (a^{2} - (b^{2} + c^{2} - 2 b c)) \\ = - (a^{2} - {(b + c)}^{2}) (a^{2} - {(b - c)}^{2}) \\ = - (a - b - c) (a + b + c) (a - b + c) (a + b - c) \\ = (a + b + c) (- a + b + c) (a - b + c) (a + b - c) \end{array}$

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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

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