[演習問題] 1-3. 因数分解 [全88問]

解答1．
共通因数をくくり出す問題です．

(1)
-2a^2 - 2ab - 2ca \\ = -2a \times a + (-2a) \times b + (-2a) \times c \\ = -2a(a + b + c)

(2)
3x^2 - 6xy + 3xz \\ =3x \times x + 3x \times (-2y) + 3x \times z \\ = 3x(x -2y + z)

(3)
6a^4b - 2a^3b^2 + 14a^2b^3 = 2a^2b(3a^2 - ab + 7b^2)

(4)
24x^3y + 144x^2y^2 - 36xy^3 = 12xy(2x^2 + 12xy -3y^2)

(5)
22a^2bc - 121ab^2c - 33abc^2 - 44abcd = 11abc(2a - 11b - 3c - 4d)

解答2．
　2次の因数分解の公式を使う練習問題です．

(1) 4a^2 + 4ab + b^2 = (2a + b)^2
(2) 9s^2 + 12st + 4t^2 = (3s + 2t)^2
(3) 144x^2 + 168xy + 49y^2 = (12x + 7y)^2
(4) x^2 + 10xy + 25y^2 = (x + 5y)^2
(5) a^2 - 4ab + 4b^2 = (a - 2b)^2
(6) 4a^2 - 12ab + 9b^2 = (2a - 3b)^2
(7) 121x^2 - 154xy + 49y^2 = (11x - 7y)^2
(8) 25x^2 - 30xy +9y^2 = (5x - 3y)^2
(9) a^2 - 4b^2 = (a + 2b)(a - 2b)
(10) 16a^2 - 9b^2 = (4a + 3b)(4a - 3b)
(11) 49x^2 - 25y^2 = (7x + 5y)(7x - 5y)
(12) 9x^2 - 64y^2 = (3x + 8y)(3x - 8y)
(13) 121a^2 + 110ab + 25b^2 = (11a + 5b)^2
(14) a^2 - 18ab + 81b^2 = (a - 9b)^2

解答3．
a^2 + b^2 + c^2 + 2ab + 2bc + 2ca \\ = a^2 + (2ab + 2ca) + (b^2 + 2bc + c^2) \\ = a^2 + 2(b + c)a + (b + c)^2 \\ = \{ a + (b + c) \}^2 \\ = (a + b + c)^2

解答4．

(1) 4a^2 + b^2 + c^2 + 4ab + 2bc + 4ca = (2a + b + c)^2
(2) a^2 + 9b^2 + 4c^2 + 6ab + 12bc + 4ca = (a + 3b + 2c)^2
(3) 4x^2 + 9y^2 + z^2 - 12xy - 6yz + 4zx = (2x - 3y + z)^2
(4) x^2 + 25y^2 + 36z^2 - 10xy + 60yz - 12zx = (x - 5y - 6z)^2
(5) 9x^2 + 49y^2 + 16z^2 + 42xy - 56yz - 24zx = (3x + 7y - 4z)^2

解答5．
　3次の公式を使う問題です．

(1) 8a^3 + 12a^2b + 6ab^2 + b^3 = (2a + b)^3
(2) 27s^3 + 54s^2t + 36st^2 + 8t^3 = (3s + 2t)^3
(3) 8x^3 + 36 x^2y + 54xy^2 + 27y^3 = (2x + 3y)^3
(4) 64x^3 + 240x^2y + 300xy^2 + 125y^3 = (4x + 5y)^3
(5) a^3 - 6a^2b + 12ab^2 - 8b^3 = (a - 2b)^3
(6) 125x^3 - 225x^2y + 135xy^2 -27y^3 = (5x - 3y)^3
(7) 8x^3 - 84x^2 + 294x - 343 = (2x - 7)^3
(8) x^3 - 9x^2 + 27x + 27 = (x - 3)^3
(9) a^3 - 8b^3 = (a - 2b)(a^2 + 2ab + 4b^2)
(10) 64a^3 + 27b^3 = (4a + 3b)(16a^2 -12ab + 9b^2)
(11) x^3 + 125y^3 = (x + 5y)(x^2 -5xy + 25y^2)
(12) 27x^3 + 8 = (3x + 2)(9x^2 - 6x + 4)
(13) x^3 - 8 = (x - 2)(x^2 + 2x + 4)
(14) x^3 - 64y^3 = (x - 4y)(x^2 + 4xy + 16y^2)
(15) 8a^3 + b^3 = (2a + b)(4a^2 - 2ab + b^2)
(16) 8a^3 - 27 = (2a - 3)(4a^2 + 6a + 9)
(17) x^3 + 9x^2 + 27x + 27 = (x + 3)^3
(18) 1331a^3 + 1815a^2b + 825ab^2 + 125b^3 = (11a + 5b)^3
(19) a^3 + b^3 -3ab + 1 = (a + b + 1)(a^2 + b^2 - ab - a - b + 1)
(20) 4a^2 - 28ab + 49b^2= (2a - 7b)^2

(20)は2次の因数分解です．

解答6．
　解法はいくつかあるので，ここではそのうち１つを示す．

(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 \\ (a + b)^3 = a^3 + b^3 + 3ab(a + b)

よって，
a^3 + b^3 = (a + b)^3 - 3ab(a + b)
この式を利用して

a^3 + b^3 + c^3 - 3abc \\ = (a + b)^3 - 3ab(a + b) + c^3 -3abc \\ = (a + b)^3 + c^3 - 3ab(a + b) -3abc \\ = \{ (a + b) + c \}^3 -3(a + b)c \{ (a + b) + c \} - 3ab(a + b) -3abc \\ = (a + b + c)^3 -3(a + b)c(a + b + c) - 3ab(a + b + c) \\ = (a + b + c) \{ (a + b + c)^2 - 3(a + b)c - 3ab \} \\ = (a + b + c)(a^2 + b^2 + c^2 + 2ab + 2bc + 2ca - 3ca - 3bc - 3ab) \\ = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca)

解答7．

(1) x^2 + 3x + 2 = (x + 1)(x + 2)
(2) x^2 - 4x + 3 = (x - 1)(x - 3)
(3) x^2 - 2x - 3 = (x + 1)(x - 3)
(4) x^2 + 8x + 15 = (x + 3)(x + 5)
(5) x^2 + 10x + 21 = (x + 3)(x + 7)
(6) x^2 + xy - 2y^2 = (x + 2y)(x - y)
(7) x^2 - 8xy + 15y^2 = (x - 3y)(x - 5y)
(8) x^2 - xy - 12y^2 = (x + 3y)(x - 4y)
(9) a^2 - ab - 42b^2 = (a + 6b)(a - 7b)
(10) a^2 + 2ab - 3b^2 = (a + 3b)(a - b)
(11) s^2 - 3st - 28t^2 = (s + 4t)(s - 7t)
(12) a^2 + 3ab - 54b^2 = (a + 9b)(a - 6b)
(13) 2x^2 - 5x + 2 = (2x - 1)(x - 2)
(14) 3x^2 - 8x - 3 = (3x + 1)(x - 3)
(15) 2x^2 + 3x + 1 = (2x + 1)(x + 1)
(16) 6x^2 - 5x + 1 = (3x - 1)(2x - 1)
(17) 6x^2 + 7x + 2 = (3x + 2)(2x + 1)
(18) 12x^2 +17x + 6 = (4x + 3)(3x + 2)
(19) 20x^2 - 13x - 15 = (5x + 3)(4x - 5)
(20) 18x^2 - 9x - 2 = (6x + 1)(3x - 2)
(21) 28x^2 + 17x - 3 = (7x - 1)(4x + 3)
(22) 6x^2 -xy - 12y^2 = (2x - 3y)(3x + 4y)
(23) x^2 - xy - 2y^2 = (x + y)(x - 2y)
(24) 2a^2 + 3x - 2 = (a + 2)(2a - 1)
(25) 2a^2 + 3ab + b^2 = (2a + b)(a + b)
(26) 12a^2 - 7ab + b^2 = (4a - b)(3a - b)
(27) 2s^2 + st - t^2 = (s + t)(2s - t)
(28) 42x^2 + 5x - 25 = (7x - 5)(6x + 5)
(29) 6x^2 -11xy - 10 y^2 = (3x + 2y)(2x - 5y)
(30) 72x^2 - 7xy - 49y^2 = (9x + 7y)(8x - 7y)

解答8．

(1)
x^2 + 5xy - x + 6y^2 -5y - 6 \\ = x^2 + (5y - 1)x + (2y - 3)(3y + 2) \\ = (x + 2y - 3)(x + 3y + 2)

(2)
2x^2 - xy + 5x - 6y^2 + 11y - 3 \\ = 2x^2 - (y - 5)x - (6y^2 - 11y + 3) \\ = 2x^2 - (y - 5)x - (3y - 1)(2y - 3) \\ = (2x + 3y - 1) \{ x - (2y - 3) \} \\ = (2x + 3y - 1)(x - 2y + 3)

(3)
6x^2 - 7xy + 7x + 2y^2 - 5y - 3 \\ = 6x^2 - 7(y - 1)x + (y - 3)(2y + 1) \\ = \{ 2x - (y - 3) \} \{ 3x - (2y + 1) \} \\ = (2x - y + 3)(x - 2y - 1)

(4)
12x^2 + 11xy + 17x + 2y^2 + 8y + 6 \\ = 12x^2 + (11y + 17)x + 2(y^2 + 4y + 3) \\ = \{ 3x + 2(y + 1) \} (4x + y + 3) \\ = (3x + 2y + 2)(4x + y + 3)

解答9．

(1)
x^4 - 81\\ = (x^2 + 9)(x^2 - 9) \\ = (x^2 + 9)(x + 3)(x - 3)

(2)
x^4 + 3x^2 + 4 \\ = (x^4 + 4x^2 + 4) - x^2 \\ = (x^2 + 2)^2 - x^2 \\ (x^2 + x + 2)(x^2 - x + 2)

(3)
x^6 - 64 \\ = (x^3 + 8)(x^3 - 8) \\ = (x + 2)(x - 2)(x^2 + 2x + 4)(x^2 - 2x + 4)

(4)
x^4 - 13x^2 + 36 \\ = (x^2 - 9)(x^2 - 4) \\ = (x + 3)(x - 3)(x + 2)(x - 2)

(5)
最も次数の低い c で式をまとめましょう．
a^2 + b^2 + 2ab - bc - ca \\ = a^2 + b^2 + 2ab - c(a + b) \\ = (a + b)^2 - c(a + b) \\ =(a + b)(a + b - c)

(6)
a^2b + ab^2 + b^2c + bc^2 + c^2a + ca^2 + 2abc \\ = a^2(b + c) + a(b^2 + 2bc + c^2) + b^2c + bc^2 \\ = a^2(b + c) + a(b + c)^2 + bc(b + c) \\ = (b + c) \{ a^2 + a(b + c) + bc \} \\ = (b + c)(a + b)(a + c) \\ = (a + b)(b + c)(c + a)

(7)
(x^2 + 9x + 18)(x^2 + 5x + 4) + 8 \\ = (x + 6)(x + 3)(x + 4)(x + 1) + 8 \\ = (x + 1)(x + 6)(x + 3)(x + 4) + 8 \\ = (x^2 + 7x + 6)(x^2 + 7x + 12) + 8 \\
ここで X = x^2 + 7x とおいて，
(x^2 + 7x + 6)(x^2 + 7x + 12) + 8 \\ = (X + 6)(X + 12) + 8 \\ = X^2 + 18X + 72 + 8 \\ = X^2 + 18X + 80 \\ = (X + 10)(X + 8) \\ = (x^2 + 7x + 10)(x^2 + 7x + 8)

(8)
「出会った中で一番難しい因数分解」といった趣旨の動画を視聴．あっさり解けてしまったので，同様の原理でもっと難しい問題を作問してみたところ，手に負えなくなりました．
x^4y^2 - x^2y^4 + x^3y + xy^3 - x^3 + y^3 + x^2y + xy^2 + xy - x + y - 1 \\ = (x^3 - x^2y + x + 1)(y^3 + xy^2 + y - 1)