【高校数学Ⅰ】1-1-4 因数分解｜問題集
1.次の式を因数分解しなさい。
(1)\(12x^3-8x^2y\)
\(=4x^2(3x-2y)\)
(2)\(3a^2x+6ax^2+ax\)
\(=ax(3a+6x+1)\)
(3)\((a+b)c+d(a+b)\)
\(=(a+b)(c+d)\)
(4)\((x-2y)a+(2y-x)b\)
\(=(x-2y)a-(x-2y)b\)
\(=(x-2y)(a-b)\)
\(=(x-2y)(a-b)\)
(5)\(4x^2y+6xy^3\)
\(=2xy(2x+3y^2)\)
(6)\(3ax^2-6a^2b\)
\(=3a(x^2-2ab)\)
(7)\(mn^2+m\)
\(=m(n^2+1)\)
(8)\(-2x^2y-6xy^2\)
\(=-2xy(x+3y)\)
(9)\((a+b)x+y(a+b)\)
\(=(a+b)(x+y)\)
(10)\((a-3b)c+(3b-a)d\)
\(=(a-3b)c-(a-3b)d\)
\(=(a-3b)(c-d)\)
\(=(a-3b)(c-d)\)
(11)\(5ab-2ac\)
\(=a(5b-2c)\)
(12)\(6a^2x-2a^2y-4az\)
\(=2a(3ax-ay-2z)\)
(13)\((a-3)+2(a-3)^2\)
\(=(a-3)\{1+2(a-3)\}\)
\(=(a-3)(2a-5)\)
\(=(a-3)(2a-5)\)
(14)\(x(y-2)+y-2\)
\(=(x+1)(y-2)\)
(15)\(a^2+6a+9\)
\(=(a+3)^2\)
(16)\(4x^2-20xy+25y^2\)
\(=(2x-5y)^2\)
(17)\(x^2-49\)
\(=(x+7)(x-7)\)
(18)\(x^2+5x+6\)
\(=(x+2)(x+3)\)
(19)\(a^2-ab-12b^2\)
\(=(a+3b)(a-4b)\)
(20)\(16a^2+8a+1\)
\(=(4a+1)^2\)
(21)\(\displaystyle x^2-x+\frac{1}{4}\)
\(\displaystyle =\left(x-\frac{1}{2}\right)^2\)
(22)\(64x^2-25y^2\)
\(=(8x+5y)(8x-5y)\)
(23)\(a^2+3ab-10b^2\)
\(=(a+5b)(a-2b)\)
(24)\(3x^2-12\)
\(=3(x^2-4)\)
\(=3(x+2)(x-2)\)
\(=3(x+2)(x-2)\)
(25)\(x^2+10x+25\)
\(=(x+5)^2\)
(26)\(x^2-12x+36\)
\(=(x-6)^2\)
(27)\(x^2+6xy+9y^2\)
\(=(x+3y)^2\)
(28)\(4a^2-4ab+b^2\)
\(=(2a-b)^2\)
(29)\(16a^2-25b^2\)
\(=(4a+5b)(4a-5b)\)
(30)\(2x^2-18y^2\)
\(=2(x^2-9y^2)\)
\(=2(x+3y)(x-3y)\)
\(=2(x+3y)(x-3y)\)
(31)\(x^2+8x+12\)
\(=(x+2)(x+6)\)
(32)\(x^2-7x+12\)
\(=(x-3)(x-4)\)
(33)\(a^2-a-20\)
\(=(a+4)(a-5)\)
(34)\(x^2+5xy+6y^2\)
\(=(x+2y)(x+3y)\)
(35)\(a^2-6ab+8b^2\)
\(=(a-2b)(a-4b)\)
(36)\(x^2-ax-12a^2\)
\(=(x+3a)(x-4a)\)
(37)\(x^3-1\)
\(=(x-1)(x^2+x+1)\)
(38)\(x^3+27a^3\)
\(=(x+3a)(x^2-3ax+9a^2)\)
(39)\(x^3-64\)
\(=(x-4)(x^2+4x+16)\)
(40)\(125x^3-y^3\)
\(=(5x-y)(25x^2+5xy+y^2)\)
(41)\(x^3+8\)
\(=(x+2)(x^2-2x+4)\)
(42)\(27x^3-125y^3\)
\(=(3x-5y)(9x^2+15xy+25y^2)\)
(43)\(1-a^3\)
\(=(1-a)(1+a+a^2)\)
(44)\(1000x^3+y^3\)
\(=(10x+y)(100x^2-10xy+y^2)\)
(45)\(3x^2+7x+2\)
\(=(3x+1)(x+2)\)
(46)\(2x^2+9x+10\)
\(=(2x+5)(x+2)\)
(47)\(2x^2-13x+6\)
\(=(2x-1)(x-6)\)
(48)\(4y^2+5y-21\)
\(=(4y-7)(y+3)\)
(49)\(3x^2+5xy-2y^2\)
\(=(3x-y)(x+2y)\)
(50)\(6x^2-7ax-3a^2\)
\(=(3x+a)(2x-3a)\)
(51)\(2x^2+5x-3\)
\(=(x+3)(2x-1)\)
(52)\(6x^2-5xy-4y^2\)
\(=(2x+y)(3x-4y)\)
(53)\(3x^2+5x-2\)
\(=(x+2)(3x-1)\)
(54)\(4a^2+8a+3\)
\(=(2a+1)(2a+3)\)
(55)\(6x^2+xy-2y^2\)
\(=(2x-y)(3x+2y)\)
(56)\(8a^2-14ab-15b^2\)
\(=(2a-5b)(4a+3b)\)
(57)\(9a^2-42ab+49b^2\)
\(=(3a-7b)^2\)
(58)\(abx^2+(a-b)x-1\)
\(=(ax-1)(bx+1)\)
(59)\(2x^2-7x+3\)
\(=(2x-1)(x-3)\)
(60)\(6a^2-ab-b^2\)
\(=(3a+b)(2a-b)\)
(61)\(abx^2+(2a^2-b^2)x-2ab\)
\(=(ax-b)(bx+2a)\)
(62)\((x-y)^2-5(x-y)+6\)
\(=(x-y-3)(x-y-2)\)
(63)\(2(x+3y)^2-(x+3y)-1\)
\(=\{2(x+3y)+1\}(x+3y-1)\)
\(=(2x+6y+1)(x+3y-1)\)
\(=(2x+6y+1)(x+3y-1)\)
(64)\((x+y)^2-9\)
\(=(x+y+3)(x+y-3)\)
(65)\(x^2-(y-1)^2\)
\(=\{(x+(y-1))\}\{x-(y-1)\}\)
\(=(x+y-1)(x-y+1)\)
\(=(x+y-1)(x-y+1)\)
(66)\(x^4-8x^2-9\)
\(=(x^2+1)(x^2-9)\)
\(=(x^2+1)(x+3)(x-3)\)
\(=(x^2+1)(x+3)(x-3)\)
(67)\(x^4-16\)
\(=(x^2+4)(x^2-4)\)
\(=(x^2+4)(x+2)(x-2)\)
\(=(x^2+4)(x+2)(x-2)\)
(68)\((x+y)^2+2(x+y)-8\)
\(=(x+y+4)(x+y-2)\)
(69)\(a-1+b(1-a)\)
\(=(a-1)-b(a-1)\)
\(=(a-1)(1-b)\)
\(=(a-1)(1-b)\)
(70)\(x^2+2xy+y^2-4\)
\(=(x+y)^2-4\)
\(=(x+y+2)(x+y-2)\)
\(=(x+y+2)(x+y-2)\)
(71)\((a-3b)(a-3b+3)-10\)
\(=(a-3b)^2+3(a-3b)-10\)
\(=(a-3b+5)(a-3b-2)\)
\(=(a-3b+5)(a-3b-2)\)
(72)\((x^2+x)^2-8(x^2+x)+12\)
\(=(x^2+x-6)(x^2+x-2)\)
\(=(x+3)(x-2)(x+2)(x-1)\)
\(=(x+3)(x-2)(x+2)(x-1)\)
(73)\(a^4-16b^4\)
\(=(a^2+4b^2)(a^2-4b^2)\)
\(=(a^2+4b^2)(a+2b)(a-2b)\)
\(=(a^2+4b^2)(a+2b)(a-2b)\)
(74)\(x^4-10x^2+9\)
\(=(x^2-9)(x^2-1)\)
\(=(x+3)(x-3)(x+1)(x-1)\)
\(=(x+3)(x-3)(x+1)(x-1)\)
(75)\(x^4+x^2+1\)
\(=x^4+2x^2+1-x^2\)
\(=(x^2+1)^2-x^2\)
\(=(x^2+x+1)(x^2-x+1)\)
\(=(x^2+1)^2-x^2\)
\(=(x^2+x+1)(x^2-x+1)\)
(76)\(x^4-1\)
\(=(x^2+1)(x^2-1)\)
\(=(x^2+1)(x+1)(x-1)\)
\(=(x^2+1)(x+1)(x-1)\)
(77)\(x^4-2x^2-8\)
\(=(x^2+2)(x^2-4)\)
\(=(x^2+2)(x+2)(x-2)\)
\(=(x^2+2)(x+2)(x-2)\)
(78)\(x^4+4\)
\(=x^4+4x^2+4-4x^2\)
\(=(x^2+2)^2-4x^2\)
\(=(x^2+2x+2)(x^2-2x+2)\)
\(=(x^2+2)^2-4x^2\)
\(=(x^2+2x+2)(x^2-2x+2)\)
(79)\(x^4-3x^2+1\)
\(=x^4-2x^2+1-x^2\)
\(=(x^2-1)^2-x^2\)
\(=(x^2+x-1)(x^2-x-1)\)
\(=(x^2-1)^2-x^2\)
\(=(x^2+x-1)(x^2-x-1)\)
(80)\(1+2ab+a+2b\)
\(=a(2b+1)+2b+1\)
\(=(a+1)(2b+1)\)
\(=(a+1)(2b+1)\)
(81)\(a^2b+a-b-1\)
\(=b(a^2-1)+a-1\)
\(=b(a+1)(a-1)+(a-1)\)
\(=(a-1)\{b(a+1)+1\}\)
\(=(a-1)(ab+b+1)\)
\(=b(a+1)(a-1)+(a-1)\)
\(=(a-1)\{b(a+1)+1\}\)
\(=(a-1)(ab+b+1)\)
(82)\(x^2+3xy+2y^2-2x-3y+1\)
\(=x^2+(3y-2)x+2y^2-3y+1\)
\(=x^2+(3y-2)x+(2y-1)(y-1)\)
\(=(x+2y-1)(x+y-1)\)
\(=x^2+(3y-2)x+(2y-1)(y-1)\)
\(=(x+2y-1)(x+y-1)\)
(83)\(x^2-xy-6y^2+2x+19y-15\)
\(=x^2-(y-2)x-(6y^2-19y+15)\)
\(=x^2-(y-2)x-(2y-3)(3y-5)\)
\(=(x+2y-3)(x-3y+5)\)
\(=x^2-(y-2)x-(2y-3)(3y-5)\)
\(=(x+2y-3)(x-3y+5)\)
(84)\(ab(a-b)+bc(b-c)+ca(c-a)\)
\(=a^2b-ab^2+b^2c-bc^2+c^2a-ca^2\)
\(=(b-c)a^2-(b^2-c^2)a+b^2c-bc^2\)
\(=(b-c)a^2-(b+c)(b-c)a+bc(b-c)\)
\(=(b-c)\{a^2-(b+c)a+bc\}\)
\(=(b-c)(a-b)(a-c)\)
\(=(b-c)a^2-(b^2-c^2)a+b^2c-bc^2\)
\(=(b-c)a^2-(b+c)(b-c)a+bc(b-c)\)
\(=(b-c)\{a^2-(b+c)a+bc\}\)
\(=(b-c)(a-b)(a-c)\)
(85)\(a^2+ab-2b-4\)
\(=(a-2)b+a^2-4\)
\(=(a-2)b+(a+2)(a-2)\)
\(=(a-2)(a+b+2)\)
\(=(a-2)b+(a+2)(a-2)\)
\(=(a-2)(a+b+2)\)
(86)\(x^2+xy-2y^2+2x-5y-3\)
\(=x^2+(y+2)x-(2y^2+5y+3)\)
\(=x^2+(y+2)x-(2y+3)(y+1)\)
\(=\{x-(y+1)\}\{x+(2y+3)\}\)
\(=(x-y-1)(x+2y+3)\)
\(=x^2+(y+2)x-(2y+3)(y+1)\)
\(=\{x-(y+1)\}\{x+(2y+3)\}\)
\(=(x-y-1)(x+2y+3)\)
(87)\((x+y+1)(x+y+2)-6\)
\(=(x+y)^2+3(x+y)+2-6\)
\(=(x+y)^2+3(x+y)-4\)
\(=(x+y+4)(x+y-1)\)
\(=(x+y)^2+3(x+y)-4\)
\(=(x+y+4)(x+y-1)\)
(88)\(4a^2-9b^2+6bc-c^2\)
\(=4a^2-(3b-c)^2\)
\(=(2a+3b-c)(2a-3b+c)\)
\(=(2a+3b-c)(2a-3b+c)\)
(89)\(x^2-2xy+3x-4y+2\)
\(=(-2x-4)y+x^2+3x+2\)
\(=-2(x+2)y+(x+1)(x+2)\)
\(=(x+2)(x-2y+1)\)
\(=-2(x+2)y+(x+1)(x+2)\)
\(=(x+2)(x-2y+1)\)
(90)\(6ab+4a-3b-2\)
\(=2a(3b+2)-(3b+2)\)
\(=(2a-1)(3b+2)\)
\(=(2a-1)(3b+2)\)
(91)\(2x^2+3xy-2y^2+x+2y\)
\(=2x^2+(3y+1)x-2y(y-1)\)
\(=(x+2y)\{2x-(y-1)\}\)
\(=(x+2y)(2x-y+1)\)
\(=(x+2y)\{2x-(y-1)\}\)
\(=(x+2y)(2x-y+1)\)
(92)\(2x^2+7xy+3y^2-5x-10y+3\)
\(=2x^2+(7y-5)x+3y^2-10y+3\)
\(=2x^2+(7y-5)x+(y-3)(3y-1)\)
\(=(x+3y-1)(2x+y-3)\)
\(=2x^2+(7y-5)x+(y-3)(3y-1)\)
\(=(x+3y-1)(2x+y-3)\)
(93)\((x-1)(x-3)(x-5)(x-7)+15\)
\(=(x-1)(x-7)(x-3)(x-5)+15\)
\(=(x^2-8x+7)(x^2-8x+15)+15\)
\(=(x^2-8x)^2+22(x^2-8x)+105+15\)
\(=(x^2-8x)^2+22(x^2-8x)+120\)
\(=(x^2-8x+10)(x^2-8x+12)\)
\(=(x^2-8x+10)(x-2)(x-6)\)
\(=(x^2-8x+7)(x^2-8x+15)+15\)
\(=(x^2-8x)^2+22(x^2-8x)+105+15\)
\(=(x^2-8x)^2+22(x^2-8x)+120\)
\(=(x^2-8x+10)(x^2-8x+12)\)
\(=(x^2-8x+10)(x-2)(x-6)\)
(94)\(a^2+2ab+b^2-c^2\)
\(=(a+b)^2-c^2\)
\(=(a+b+c)(a+b-c)\)
\(=(a+b+c)(a+b-c)\)
(95)\(x^2-y^2-zx+yz\)
\(=(x+y)(x-y)-z(x-y)\)
\(=(x-y)(x+y-z)\)
\(=(x-y)(x+y-z)\)
(96)\(2x^2+5xy+2y^2-x+y-1\)
\(=2x^2+(5y-1)x+2y^2+y-1\)
\(=2x^2+(5y-1)x+(2y-1)(y+1)\)
\(=(2x+y+1)(x+2y-1)\)
\(=2x^2+(5y-1)x+(2y-1)(y+1)\)
\(=(2x+y+1)(x+2y-1)\)
(97)\(a(b^2-c^2)+b(c^2-a^2)+c(a^2-b^2)\)
\(=a(b^2-c^2)+bc^2-a^2b+ca^2-b^2c\)
\(=(c-b)a^2+(b^2-c^2)a+bc^2-bc^2\)
\(=(c-b)a^2-(c-b)(b+c)a+bc(c-b)\)
\(=(c-b)\{a^2-(b+c)a+bc\}\)
\(=(c-b)(a-b)(a-c)\)
\(=(c-b)a^2+(b^2-c^2)a+bc^2-bc^2\)
\(=(c-b)a^2-(c-b)(b+c)a+bc(c-b)\)
\(=(c-b)\{a^2-(b+c)a+bc\}\)
\(=(c-b)(a-b)(a-c)\)
(98)\(a^2(b+c)+b^2(c+a)+c^2(a+b)+2abc\)
\(=a^2(b+c)+b^2c+b^2a+c^2a+c^2b+2abc\)
\(=(b+c)a^2+(b^2+2bc+c^2)a+b^2c+bc^2\)
\(=(b+c)a^2+(b+c)^2a+bc(b+c)\)
\(=(b+c)\{a^2+(b+c)a+bc\}\)
\(=(b+c)(a+b)(a+c)\)
\(=(b+c)a^2+(b^2+2bc+c^2)a+b^2c+bc^2\)
\(=(b+c)a^2+(b+c)^2a+bc(b+c)\)
\(=(b+c)\{a^2+(b+c)a+bc\}\)
\(=(b+c)(a+b)(a+c)\)
次の学習に進もう！