因数定理
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f(a)=0ならf(x)は因数(x-a)を持つ。 |
多項式f(x)
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因数分解結果
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f(x)=x3 -6x2
+9x-4
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因数分解：f(1)=f(4)=0,
f(2)=8-24+18-4=-2,f(-2)<0
f(x)=(x-1)(x-4)(x-1)=(x-4)(x-1)2 |
f(x)=16x4 +96x3
+176x2 +96x-9
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(2x+3)2 (4x2
+12x-1)
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| (x2-y2)2-8(x2+y2)+16
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=(x+y)2 (x-y)2-4{(x+y)2
+(x-y)2)}+16
={(x+y)2-4}{ (x-y)2 -4}
=(x+y-2)(x+y+2)(x-y+2)(x-y-2) |
| f(a,b,c)=2ab(a-b)-bc(b+2c)+2ac(2a+c)-3abc |
f(a,a,c)=-ac(a+2c)+2ac(2a+c)-3a2c
=-a2c+4a2c-3a2c-2ac2+2ac2=0
f(a,b,c)=2ab(a-b)+2c2(a-b)+c(a2-b2)+3ac(a-b)}
=(a-b){2ab+2c2 +c(a+b)+3ac}
=(a-b)(2ab+2c2 +cb+4ac)=(a-b)(b+2c)(2a+c)
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f(a,b,c)=(a+b)(b+c)(c+a)+abc
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=(b+c){a2+(b+c)a+bc}+abc
=(b+c)a2+{(b+c)2+bc}a+bc(b+c)
=(a+b+c){(b+c)a+bc}=(a+b+c)(ab+bc+ca)
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剰余定理
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整式f(x)をp(x)で割ったときの商をq(x)、余りをr(x)とすれば、r(x)の次数は
p(x)の次数から１を引いた次数以下であり、f(x)=p(x)q(x)+r(x) と書ける。p(a)=0の時
f(a)=r(a)となる。
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因数分解公式
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x2+(a+b)x+ab
x2-(a+b)x+ab
x2+2ax+a2
x2-2ax+a2
x2-a2
a2-b2
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(x+a)(x+b)
(x-a)(x-b)
(x+a)2
(x-a)2
(x+a)(x-a)
(a+b)(a-b)
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a2±2ab+b2=(a±b)2
(複合同順)
a3±3a2b+3ab2±b3=(a±b)3
(複合同順)
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acx2±(ad+bc)x+bd=(ax±b)(cx±d)
(複合同順)
a4+a2b2+b4=(a2+ab+b2)
(a2-ab+b2)
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3x2+14x+8=(x＋4)(3x＋2)
2x2-5x+2= (2x-1)(x-2)
x2-xy+6y2=(x+2y)(x-3y)
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(xn±1)
の因数分解
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xn-1 =(x-1)(xn-1+xn-2+
… +x+1) (n≧2)
x2n+1+1 =(x+1)(x2n-x2n-1+
… -x+1) (n≧1) |
x2-1
x3-1
x3+1
x4-1
x4+1
x5-1
x5+1
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(x-1)(x+1)
(x-1)(x2+x+1)
(x+1)(x2-x+1)
(x-1)(x+1)(x2+1)
(x2-√2x+1)(x2+√2x+1)
(x-1)(x4+x3+x2+x+1)
(x+1)(x4-x3+x2-x+1)
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x6-1
x6+1
x7-1
x7+1
x8-1
x8+1
x9-1
x9+1
x10-1
x10+1 |
(x-1)(x+1)(x4+x2+1)=(x-1)(x+1)(x2+x+1)(x2-x+1)
(x2+1)(x4-x2+1)=(x2+1)(x2-√3x+1)(x2+√3x+1)
(x-1)(x6+x5+x4+x3+x2+x+1)
(x+1)(x6-x5+x4-x3+x2-x+1)
(x-1)(x+1)(x2+1)(x2-√2x+1)(x2+√2x+1)
(x^4-√2x2+1)(x4+√2x2+1)
=(x2+1+x√(2+√2))(x2+1-x√(2+√2))(x2+1+x√(2-√2))(x2+1-x√(2-√2))
(x-1)(x2+x+1)(x6+x3+1)
(x+1)(x2-x+1)(x6-x3+1)
(x-1)(x+1)(x4+x3+x2+x+1)(x^4-x3+x2-x+1)
(x2+1)(x8-x6+x4-x2+1)
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x11-1
x11+1
x12-1
x12+1
x13-1
x13+1
x14-1
x14+1
x15-1
x15+1
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(x-1)(x10+x9+x8+x7+x6+x5+x4+x3+x2+x+1)
(x+1)(x10-x9+x8-x7+x6-x5+x4-x3+x2-x+1)
(x-1)(x+1)(x2+1)(x2+x+1)(x2-x+1)(x2-√3x+1)(x2+√3x+1)
(x2-√2x+1)(x2+√2x+1)(x4-√3x2+1)(x4+√3x2+1)
=(x2-√2x+1)(x2+√2x+1)(x2+1+x√(2+√3))(x2+1+x√(2-√3))
(x-1)(x12+x11+x10+x9+x8+x7+x6+x5+x4+x3+x2+x+1)
(x+1)(x12-x11+x10-x9+x8-x7+x6-x5+x4-x3+x2-x+1)
(x-1)(x+1)(x6+x5+x4+x3+x2+x+1)(x6-x5+x4-x3+x2-x+1)
(x2+1)(x12-x10+x8-x6+x4-x2+1)
(x-1)(x2+x+1)(x4+x3+x2+x+1)(x8-x7+x5-x4+x3-x+1)
(x+1)(x2-x+1)(x4-x3+x2-x+1)(x8+x7-x5-x4-x3+x+1)
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an-bn= (a-b)(a2+ab+b2)
a2n+1+b2n+1= (a+b)(a2-ab+b2)
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(a-b)(an-1+an-2b+ …
+abn-2+bn-1) (n≧2)
(a+b)(a2n-a2n-1b+ … -ab2n-1+b2n)
(n≧1) |
a3+b3=
(a+b)(a2-ab+b2)
a3-b3= (a-b)(a2+ab+b2)
a3+(a+b)3= (2a+b)(a2+ab+b2)
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(x2-y2)2-8(x2+y2)+16=(x+y+2)(x+y-2)(x-y+2)(x-y-2)
(x2-1)(y2-1)-4xy=(xy-y-x-1)(xy+y+x-1)
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a(b-c)3+b(c-a)3+c(a-b)3
=(a-b)(b-c)(c-a)(a+b+c)
a(b3-c3)+b(c3-a3)+c(a3-b3)
=(a-b)(b-c)(c-a)(a+b+c)
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x3+(2a+1)x2+(a2+2a-1)x+a2-1=(x+1)(x+a+1)(x+a-1) |
a4+b4+c4-2b2c2-2c2a2-2a2b2
=-(a+b+c)(b+c-a)(c+a-b)(a+b-c) |
x2-2xy+y2-3x+3y+2=(x-y-1)(x-y-2)
x2-6xy+9y2-2x+6y-24=(x-3y+4)(x-3y-6)
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(a+b)(b+c)(c+a)+abc
=(a+b+c)(ab+bc+ca)
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x3+y3+z3-3xyz=(x+y+z)(x2+y2+z2-xy-yz-xz)
x3+y3+z3+xy(x+y)+yz(y+z)+xz(x+z)=(x+y+z)(x2+y2+z2)
x(y2-z2)+y(z2-x2)+z(x2-y2)=
(x-y)(y-z)(z-x)
xy2-x2y+yz2-y2z+zx2-z2x=xy(y-x)+yz(z-y)+zx(x-z)=(x-y)(y-z)(z-x)
x(y3-z3)+y(z3-x3)+z(x3-y3)=(x-y)(y-z)(z-x)(x+y+z)
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