3年数学
 多項式

 こたえ  〜 3年1章の正答 〜
 

(1)多項式×単項式

3x+3y 7x−7y 2a+2b 9a−9b 4a+4
5x−20 −3x−12 x+3 12x+6 10 −6x+8
11 5x−110y 12 5x−15y 13 axay  14 x−7x 15 3x−6xy
16 4a+6ab−10a 17 10x+6 18 −2x+2a     

(2)多項式÷単項式

2x+4y 5x−3y 2a+3b a−5b 9a−6
−3x+2 2x+7 −2x+1 x+4y+5 10 3x−4x−5
11 x+3y−2 12 xy 13 x+3y  14 2x−5y+4   

(3)式の展開

ab+3a+2b+6 ab+4a+5b+20 xy6xy+6 xy+2x+8y+16
xy−3x−7y+21 xy−4x−3y+12 ab−2a+5b−10 ab3ab−3
2xy+6xy+3 10 5xy−4x−10y+8 11 xybxayab 12 acadbcbd
13 3a+10ab−8b 14 xxyy     

(4)一般型の展開 1

x2+12x+35 x2+8x+12 x2−9x+20 x2−11x+18
x2+3x−28 x2+2x−3 x2−6x−16 x2x−30
x2−3x−4 10 x2−6x−27 11 x2−14x+48 12 x2−6x−27
13 a2+4a−5 14 y2−2y−8 15 b2+11b+28 16 n2n−6

(5)一般型の展開 2

x2+5xy+6y2 x2+6xy+5y2 x2−8xy+12y2 x2−9xy+20y2
x2+2xy−8y2 x2xy−2y2 x2−2xy−3y2 a2−3ab−28b2
4x2+16x+15 10 9x2−3x−2 11 25a2−15a+2 12 4y2−6y−28

(6)平方型の展開

x2+4x+4 x2−10x+25 x2+8x+16 x2−12x+36
x2+2x+1 x2−16x+64 x2−14x+49
x2x 1
4
a2+20a+100 10 y2−18y+81 11 x2+2xyy2  12 x2−4xy+4y2
13 4x2−4x+1 14 9x2+6xyy2 15 x2−10x+25 16 4x2+28xy+49y2

(7)和差型の展開

x2−9 x2−64 x2−4 x2−25 x2−81 x2−1
x2−16
x2  1
 4
9x2−1 10 4x2−1 11 x2−25y2 12 a2−4b2
13 16−9x2  (−9x2+16) 14 x2−81 15 25a2−9b2 16
x2
16
  

(8)乗法公式

x2+7x+12 x2+4x−5 x2+2x+1 x2−49
4x2−12x x2x−56 a2+11a+30 y2−12y+32
x2 4 x 4
3 9
10 m2−0.25 11 a2bab2 12 x2xy−6y2
13 9x2−64y2 14 4a2+12a+9 15 4x2−4x−3 16 6x2−13x−5

(9)いろいろな式の展開

(x2+7x+10)+(x2+4x+3)=2x2+11x+13 (x2+9x+20)−(x2+20)=9x 
(x2+6x+9)+(x2x−2)=2x2+7x+7 (3x2−3x)−(x2−4)=2x2−3x+4
(x2−10x+25)+(x2+10x+25)=2x2+50 2(x2x−12)−(x2+8x+16)=x2−10x−40
(A+2)(A−4)=A2−2A−8
=(xy)2−2(xy)−8
x2+2xyy2−2x−2y−8
(A+3)(A−3)=A2−9
=(a−2b)2−9
a2−4ab+4b2−9

(10)共通因数

5(ab) 4(ab) 3(a+3b) 2(x−4y)
2(3a+4b) 7(2x−3y) a(xy) 2a(2x+3y)
x(x+3) 10 2x(x−3) 11 2c(3a−2b) 12 4a2(3x+4y)
13 c(5a+4b) 14 x2(x−6) 15 xy(xy) 16 5x(3yy+2)

(11)一般型の因数分解

(x+2)(x+3) (x+2)(x+4) (x+1)(x+2) (x+6)(x+9)
(x−3)(x−4) (x−5)(x−7) (x−2)(x−7) (x−3)(x−8)
(x+4)(x−2) 10 (x+7)(x−3) 11 (x+5)(x−4) 12 (x+9)(x−2)
13 (x+2)(x−7) 14 (x+4)(x−5) 15 (x+6)(x−4) 16 (x+3)(x−6)
17 (a+2)(a−8) 18 (x+9)(x−7)     

(12)平方型の因数分解

(x+2)2 (x+6)2 (x+9)2 (x+1)2 (x+3)2
(x+7)2 (x−4)2 (x−10)2 (x−8)2 10 (x−12)2
11 (2x+1)2 12 (5x−1)2 13 (2x+5)2 14 (x−3y)2  15 (x−0.5)2
16
(x 1 )2
3
17 (a+2b)2 18 (3x−2)2     

(13)和差型の因数分解

(x+6)(x−6) (x+9)(x−9) (x+3)(x−3)
(x+1)(x−1) (x+4)(x−4) (x+7)(x−7)
(a+2)(a−2) (y+10)(y−10) (5x+1)(5x−1)
10 (8xy)(8xy) 11 (3a+44b)(3a−4b) 12 (1+6x)(1−6x)
13 (x+0.2)(x−0.2) 14
(x 1 )(x 1 )
2 2
15
(a 5 b)(a 5 b)
9 9
16 (A+11)(A−11)     

(14)いろいろな因数1分解 1

2(x2+5x+6)=2(x+2)(x+3) 3(x2−8x+15)=3(x−3)(x−5) 
5(x2+6x+9)=5(x+3)2 8(x2−1)=8(x+1)(x−1)
2(x2−3x−18)=2(x−6)(x+3) a(x2+4x+4)=a(x+2)2 
3(x2−10x−11)=3(x−11)(x+1) −2(a2−5a−24)=−2(a−8)(a+3) 
−3(x2−9)=−3(x+3)(x−3) 10 2(25x2y2)=2(5xy)(5xy)
11 x(16x2y2)=x(4xy)(4xy) 12 2x(a2+3a−18)=2x(a+6)(a−3)
13 2(4y2−20y+25)=2(2y−5)2 14
y(x2x 1 )=y(x 1
4 2

(15)いろいろな因数1分解 2

2−M=M(M−1)
=(x+1)(x+1−1)
x(x+1)
2  (axay)+(xy)
a(xy)+(xy)=aM+M
=M(a+1)=(xy)(a+1) 
3  (M+3)2
=M2+6M+9
=(ab)2+6(ab)+9
a2−2abb2+6a−6b+9
 
4  y(x+1)−3(x+1)=yM−3M
=M(y−3)=(x+1)(y−3)
【別解】
 x(y−3)+(y−3)=xM+M
=M(x+1)=(y−3)(x+1)
5  2+5M+6
=(M+2)(M+3)
=(x+1+2)(x+1+3)
=(x+3)(x+4)
(a2+2a+1)−b2=(a+1)2b2
=M2b2=(M+b)(M−b)
=(a+1+b)(a+1−b)
 または,(ab+1)(ab+1)
2−M−2=(M−2)(M+1)
=(x−3−2)(x−3+1)
=(x−5)(x−2) 
(x2)2−(y2)2
=(x2y2)(x2y2)
=(x2y2)(xy)(xy)
9  y(x−1)+(x−1)
yM+M
=M(y+1)
=(x−1)(y+1)
10  x2(y−1)−(y−1)=x2M−M
=M(x2−1)=M(x+1)(x−1)
=(y−1)(x+1)(x−1)
 または,(x+1)(x−1)(y−1)

(16)式の計算の利用

[1] (100−3)=100−6×100+9=10000−600+9=9409
(100−6)(100+6)=100−6=10000−36=9964
(80+2)(80+3)=80+5×80+6=6400+400+6=6806
(77+67)(77−67)=144×10=1440
[2] (x−4)(x+2)=(14−4)(14+2)=10×16=160
(xy)2=(8−3)2=52=25
(a2−2abb2)+ab=(ab)2ab=72+18=49+18=67
3(2x+3)+2(x−4)=6x+9+2x−8=8x+1

(17)素因数分解

[1] 2 2×3 2×3 3 2×3×5
2×3 3×5 2×3 2×5 10 2×3×5
[2] 45=3×5だから,5をかければよい
120=2×3×5だから,2×3×5=30で割ればよい

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