기탄수학 M2 답지 정답

전체 글

(1)

정답및풀이

※ 정답 및 풀이는

따로 보관하고 있다가

채점할 때 사용해 주세요.

개인별∙능력별 학습 프로그램

www.gitan.co.kr/ (02)586-1007(대)

(2)

M - 61 ( 1 )

x

( 2 )

b

( 3 )

x

(

y

+

z

) ( 4 )

x

, 5 ( 5 )

a

(

b

+3

c

) ( 6 )

x

(

y

-1) ( 7 )

a

(

a

+2) ( 8 )-

x

(

x

+

y

) ( 9 )5 (10)4 (11)3(3

a

+2) (12)5(2

b

-5) (13)2

x

, 3

y

(14)3(

x

-3

y

) (15)2(4

x

+

y

) (16)5(4

a

-3

b

) (17)4(4

a

+5

b

) (18)-2(3

x

2+2

y

) <풀이> ※ 다항식의 각 항에 공통인수가 있을 때에는 분 배법칙을 이용하여 공통인수로 묶어 내어 인수 분해한다.

ma

+

mb

=

m

(

a

+

b

) ( 1 )

ax

+

ay

=

a

×

x

+

a

×

y

=

a

(

x

+

y

) ( 2 )

ab

-

bc

=

b

×

a

-

b

×

c

=

b

(

a

-

c

) ( 3 )

xy

+

xz

=

x

×

y

+

x

×

z

=

x

(

y

+

z

) ( 4 )-

cx

+5

c

=(-

c

x

-(-

c

)×5=-

c

(

x

-5) <다른 풀이> -

cx

+5

c

=5

c

-

cx

=

c

×5-

c

×

x

=

c

(5-

x

) ( 5 )

ab

+3

ac

=

a

×

b

+

a

×3

c

=

a

(

b

+3

c

) ( 6 )

xy

-

x

=

x

×

y

-

x

×1=

x

(

y

-1) ( 7 )

a

2+2

a

=

a

×

a

+

a

×2=

a

(

a

+2) ( 8 )-

x

2 -

xy

=(-

x

x

+(-

x

y

=-

x

(

x

+

y

) ( 9 )2

x

+10=2×

x

+2×5=2(

x

+5) (10)4

y

-28=4×

y

-4×7=4(

y

-7) (11)9

a

+6=3×3

a

+3×2=3(3

a

+2) (12)10

b

-25=5×2

b

-5×5=5(2

b

-5) (13)-8

x

+12

y

=(-4)×2

x

-(-4)×3

y

=-4(2

x

-3

y

) (14)3

x

-9

y

=3×

x

-3×3

y

=3(

x

-3

y

) (15)8

x

+2

y

=2×4

x

+2×

y

=2(4

x

+

y

) (16)20

a

-15

b

=5×4

a

-5×3

b

=5(4

a

-3

b

) (17)16

a

+20

b

=4×4

a

+4×5

b

=4(4

a

+5

b

) (18)-6

x

2 -4

y

=(-2)×3

x

2 +(-2)×2

y

=-2(3

x

2 +2

y

) M - 63 ( 1 )2

ab

(

x

+1) ( 2 )5

xy

(

a

-1) ( 3 )-4

ax

(

b

-2) ( 4 )2

bx

(5

a

-1) ( 5 )3

xz

(2

y

+3) ( 6 )3

x

2(

a

-

b

) ( 7 )6

y

2(

x

+

z

) ( 8 )-5

xy

(

x

+2) ( 9 )3

xy

(3

y

+

x

) (10)2

xz

(2

y

2

z

-3

x

) (11)

x

2(

x

+1) (12)

x

3(

x

-

y

) (13)-

x

4(

ax

-

b

) (14)

x

2

y

2(

x

-

y

) (15)

a

2

b

2(3

a

+4

b

) (16)6

ax

2(

a

-1) (17)4

xy

2 (

y

2 +2) (18)-2

axy

(

y

+3

x

) (19)5

axy

2 (2

x

2 +

y

) (20)3

x

2

yz

2 (3

x

-2

z

) <풀이> ( 1 )2

abx

+2

ab

=2

ab

×

x

+2

ab

×1=2

ab

(

x

+1) ( 2 )5

axy

-5

xy

=5

xy

×

a

-5

xy

×1=5

xy

(

a

-1) ( 3 )-4

abx

+8

ax

=(-4

ax

b

-(-4

ax

)×2 =-4

ax

(

b

-2) M - 62 ( 1 )3

x

( 2 )5

x

(

x

-4) ( 3 )

a

, 1 ( 4 )2

x

( 5 )4

b

(2

a

+3

c

) ( 6 )4

y

(

x

-2) ( 7 )9

x

(

x

+

y

) ( 8 )-6

y

(

x

+3

y

)( 9 )3

x

(5

xy

+2) (10)2

b

(5

ab

-4)(11)

ab

(

x

+

y

) (12)

xy

(13)

x

, 3 (14)

by

(

a

-

x

) (15)

ax

(

x

+5) (16)

xy

(

y

-1) (17)

xy

(

x

+1) (18)-

xy

(

xy

+2) (19)

xy

(4

x

+

y

) (20)

ab

(

a

2 -

b

3) <풀이> ※ 분배법칙을 이용하여 전개한 식이므로 분배 법칙을 이용하여 괄호를 풀기 전의 식으로 고치 도록 한다. 또, 공통인수를 묶어 낼 때 공통인수 의 일부분만 묶어 내지 않도록 주의한다. ( 1 )9

x

2 +3

x

=3

x

×3

x

+3

x

×1=3

x

(3

x

+1) ( 2 )5

x

2-20

x

=5

x

×

x

-5

x

×4=5

x

(

x

-4) ( 3 )-7

ax

+7

x

=(-7

x

a

-(-7

x

)×1 =-7

x

(

a

-1) ( 4 )6

xy

-8

xz

=2

x

×3

y

-2

x

×4

z

=2

x

(3

y

-4

z

) ( 5 )8

ab

+12

bc

=4

b

×2

a

+4

b

×3

c

=4

b

(2

a

+3

c

) ( 6 )4

xy

-8

y

=4

y

×

x

-4

y

×2=4

y

(

x

-2) ( 7 )9

x

2 +9

xy

=9

x

×

x

+9

x

×

y

=9

x

(

x

+

y

) ( 8 )-6

xy

-18

y

2=(-6

y

x

+(-6

y

)×3

y

=-6

y

(

x

+3

y

) ( 9 )15

x

2

y

+6

x

=3

x

×5

xy

+3

x

×2=3

x

(5

xy

+2) (10)10

ab

2 -8

b

=2

b

×5

ab

-2

b

×4=2

b

(5

ab

-4) (11)

abx

+

aby

=

ab

×

x

+

ab

×

y

=

ab

(

x

+

y

) (12)

axy

-

bxy

=

xy

×

a

-

xy

×

b

=

xy

(

a

-

b

) (13)-

abx

+3

ab

=(-

ab

x

-(-

ab

)×3 =-

ab

(

x

-3) (14)

aby

-

bxy

=

by

×

a

-

by

×

x

=

by

(

a

-

x

) (15)

ax

2+5

ax

=

ax

×

x

+

ax

×5=

ax

(

x

+5) (16)

xy

2-

xy

=

xy

×

y

-

xy

×1=

xy

(

y

-1) (17)

x

2

y

+

xy

=

xy

×

x

+

xy

×1=

xy

(

x

+1) (18)-

x

2

y

2 -2

xy

=(-

xy

xy

+(-

xy

)×2 =-

xy

(

xy

+2) (19)4

x

2

y

+

xy

2 =

xy

×4

x

+

xy

×

y

=

xy

(4

x

+

y

) (20)

a

3

b

-

ab

4 =

ab

×

a

2-

ab

×

b

3=

ab

(

a

2 -

b

3)

(3)

( 4 )10

abx

-2

bx

=2

bx

×5

a

-2

bx

×1 =2

bx

(5

a

-1) ( 5 )6

xyz

+9

xz

=3

xz

×2

y

+3

xz

×3=3

xz

(2

y

+3) ( 6 )3

ax

2-3

bx

2=3

x

a

-3

x

b

=3

x

2(

a

-

b

) ( 7 )6

xy

2 +6

y

2

z

=6

y

2 ×

x

+6

y

2 ×

z

=6

y

2 (

x

+

z

) ( 8 )-5

x

2

y

-10

xy

=(-5

xy

x

+(-5

xy

)×2 =-5

xy

(

x

+2) ( 9 )9

xy

2+3

x

2

y

=3

xy

×3

y

+3

xy

×

x

=3

xy

(3

y

+

x

) (10)4

xy

2

z

2 -6

x

2

z

=2

xz

×2

y

2

z

-2

xz

×3

x

=2

xz

(2

y

2

z

-3

x

) (11)

x

3+

x

2=

x

x

+

x

2×1=

x

2(

x

+1) (12)

x

4-

x

3

y

=

x

3 ×

x

-

x

3 ×

y

=

x

3 (

x

-

y

) (13)-

ax

5+

bx

4=(-

x

4 )×

ax

-(-

x

4 )×

b

=-

x

4(

ax

-

b

) (14)

x

3

y

2-

x

2

y

3=

x

2

y

x

-

x

2

y

y

=

x

2

y

2(

x

-

y

) (15)3

a

3

b

2+4

a

2

b

3=

a

2

b

2 ×3

a

+

a

2

b

2 ×4

b

=

a

2

b

2(3

a

+4

b

) (16)6

a

2

x

2-6

ax

2=6

ax

2 ×

a

-6

ax

2 ×1 =6

ax

2(

a

-1) (17)4

xy

4+8

xy

2=4

xy

2 ×

y

2+4

xy

2 ×2 =4

xy

2(

y

2+2) (18)-2

axy

2-6

ax

2

y

=(-2

axy

y

+(-2

axy

)×3

x

=-2

axy

(

y

+3

x

) (19)10

ax

3

y

2+5

axy

3=5

axy

2 ×2

x

2+5

axy

2 ×

y

=5

axy

2(2

x

2+

y

) (20)9

x

3

yz

2-6

x

2

yz

3=3

x

2

yz

2×3

x

-3

x

2

yz

2×2

z

=3

x

2

yz

2(3

x

-2

z

) M - 64 ( 1 )

a

(

x

+

y

+

z

) ( 2 )

x

(

a

+

b

-

c

) ( 3 )-

a

(3

x

+5

y

-

z

) ( 4 )2

x

(2

a

-3

b

-

c

) ( 5 )

x

(

x

+

y

+

z

) ( 6 )-

ab

(

x

2 -

x

+

c

) ( 7 )4

x

(

x

-

y

+2) ( 8 )3

x

(2

x

-

y

-3

z

) ( 9 )

x

(

x

2+

x

+1) (10)2

x

(

x

2+3

x

-2) (11)-

xy

(

x

+

y

-2) (12)5

a

(

x

2 -3

y

2-2

z

2) (13)

xy

2(

x

+2

y

+

z

) (14)-

xyz

(

x

-

y

+

z

) (15)3

abc

(3

ab

-2

b

+4) (16)

xy

2

z

3(

xy

2-3

x

2

z

-5

yz

2) <풀이> ( 1 )

ax

+

ay

+

az

=

a

×

x

+

a

×

y

+

a

×

z

=

a

(

x

+

y

+

z

) ( 2 )

ax

+

bx

-

cx

=

x

×

a

+

x

×

b

-

x

×

c

=

x

(

a

+

b

-

c

) ( 3 )-3

ax

-5

ay

+

az

=(-

a

)×3

x

+(-

a

)×5

y

-(-

a

z

=-

a

(3

x

+5

y

-

z

) ( 4 )4

ax

-6

bx

-2

cx

=2

x

×2

a

-2

x

×3

b

-2

x

×

c

=2

x

(2

a

-3

b

-

c

) ( 5 )

x

2+

xy

+

xz

=

x

×

x

+

x

×

y

+

x

×

z

=

x

(

x

+

y

+

z

)

( 6 )-

abx

2+

abx

-

abc

=(-

ab

x

2-(-

ab

x

+(-

ab

c

=-

ab

(

x

2-

x

+

c

) ( 7 )4

x

2-4

xy

+8

x

=4

x

×

x

-4

x

×

y

+4

x

×2 =4

x

(

x

-

y

+2) ( 8 )6

x

2-3

xy

-9

xz

=3

x

×2

x

-3

x

×

y

-3

x

×3

z

=3

x

(2

x

-

y

-3

z

) ( 9 )

x

3+

x

2+

x

=

x

×

x

2+

x

×

x

+

x

×1=

x

(

x

2+

x

+1) (10)2

x

3+6

x

2-4

x

=2

x

×

x

2+2

x

×3

x

-2

x

×2 =2

x

(

x

2+3

x

-2) (11)-

x

2

y

-

xy

2+2

xy

=(-

xy

x

+(-

xy

y

-(-

xy

)×2 =-

xy

(

x

+

y

-2) (12)5

ax

2-15

ay

2-10

az

2 =5

a

×

x

2-5

a

×3

y

2-5

a

×2

z

2 =5

a

(

x

2-3

y

2-2

z

2) (13)

x

2

y

2+2

xy

3+

xy

2

z

=

xy

2 ×

x

+

xy

2 ×2

y

+

xy

2 ×

z

=

xy

2(

x

+2

y

+

z

) (14)-

x

2

yz

+

xy

2

z

-

xyz

2

=(-

xyz

x

-(-

xyz

y

+(-

xyz

z

=-

xyz

(

x

-

y

+

z

)

(15)9

a

2

b

2

c

-6

ab

2

c

+12

abc

=3

abc

×3

ab

-3

abc

×2

b

+3

abc

×4 =3

abc

(3

ab

-2

b

+4) (16)

x

2

y

4

z

3-3

x

3

y

2

z

4-5

xy

3

z

5 =

xy

2

z

3 ×

xy

2-

xy

2

z

3 ×3

x

2

z

-

xy

2

z

3 ×5

yz

2 =

xy

2

z

3(

xy

2-3

x

2

z

-5

yz

2) M - 65 ( 1 )

a

,

b

( 2 )

x

,

a

( 3 )(

x

+1)(

a

-

b

) ( 4 )(

a

-4)(

x

-

y

) ( 5 ) (

x

+

y

)(

a

+

b

) ( 6 )(

a

-

b

)(

x

+

y

) ( 7 )(

a

+3)(

x

-1) ( 8 )(

x

-5)(1-

a

) ( 9 )2

a

, 1 (10)(

a

-

b

)(1+3

x

) (11)(

x

+2)(2

x

-3) (12)(2

a

-5)(

x

-

y

) (13)(3

x

+2)(

a

+

b

) (14)(3

a

+5)(

x

-2

y

) (15)(4

x

-1)(3

a

+

b

) (16)(

x

-2

y

)(3

y

-5

z

) (17)(2

x

-3

y

)(5

a

+2

b

) (18)(4

a

+3

b

)(4

x

-3

y

)

(4)

<풀이> ※ (

a

+

b

)

x

+(

a

+

b

)

y

를 인수분해할 때, 전개하지 말고 두 다항식 (

a

+

b

)

x

와 (

a

+

b

)

y

에 공통으로 들어 있는 인수

a

+

b

로 묶어 내면 다음과 같이 인수분해할 수 있다. (

a

+

b

)

x

+(

a

+

b

)

y

=(

a

+

b

)(

x

+

y

) ( 1 )

x

(

a

+

b

)+2(

a

+

b

)=(

a

+

b

)(

x

+2) ( 2 )

x

(

x

-

y

)+

a

(

x

-

y

)=(

x

-

y

)(

x

+

a

) ( 3 )

a

(

x

+1)-

b

(

x

+1)=(

x

+1)(

a

-

b

) ( 4 )

x

(

a

-4)-

y

(

a

-4)=(

a

-4)(

x

-

y

) ( 5 )(

x

+

y

)

a

+(

x

+

y

)

b

=(

x

+

y

)(

a

+

b

) ( 6 )(

a

-

b

)

x

+(

a

-

b

)

y

=(

a

-

b

)(

x

+

y

) ( 7 )

x

(

a

+3)-(

a

+3)=(

a

+3)(

x

-1) ( 8 )(

x

-5)-

a

(

x

-5)=(

x

-5)(1-

a

) ( 9 )2

a

(

x

+

y

)+(

x

+

y

)=(

x

+

y

)(2

a

+1) (10)(

a

-

b

)+3

x

(

a

-

b

)=(

a

-

b

)(1+3

x

) (11)2

x

(

x

+2)-3(

x

+2)=(

x

+2)(2

x

-3) (12)(2

a

-5)

x

-(2

a

-5)

y

=(2

a

-5)(

x

-

y

) (13)(3

x

+2)

a

+(3

x

+2)

b

=(3

x

+2)(

a

+

b

) (14)

x

(3

a

+5)-2

y

(3

a

+5)=(3

a

+5)(

x

-2

y

) (15)3

a

(4

x

-1)+

b

(4

x

-1)=(4

x

-1)(3

a

+

b

) (16)3

y

(

x

-2

y

)-5

z

(

x

-2

y

)=(

x

-2

y

)(3

y

-5

z

) (17)5

a

(2

x

-3

y

)+2

b

(2

x

-3

y

)=(2

x

-3

y

)(5

a

+2

b

) (18)4

x

(4

a

+3

b

)-3

y

(4

a

+3

b

)=(4

a

+3

b

)(4

x

-3

y

) M - 66 ( 1 )

a

,

b

( 2 )4, 1 ( 3 )3(

x

-

y

)(

x

-3) ( 4 )2(

a

+

b

)(

x

+3

y

) ( 5 ) 2(

x

+

y

)(3

a

-4

b

) ( 6 )3(

a

-

b

)(2

a

+3

b

) ( 7 )3

b

,

a

,

c

( 8 )6

x

(

x

-

y

)(

a

+

b

) ( 9 )

x

(

x

+4)(3

x

-

y

) (10)

a

(

x

-5)(5

a

-1) (11)2

x

(

a

+

b

)(4

x

+1) (12)

xy

(

a

-1)(

x

+

y

) (13)2

xy

(

x

+

y

)(2

a

-3

b

)(14)3

x

2(

x

-

y

)(3

a

-2

b

) <풀이> ( 1 )5

a

(

x

-3)+5

b

(

x

-3)=(

x

-3)(5

a

+5

b

) =5(

x

-3)(

a

+

b

) ( 2 )4

x

(

a

+5)-4(

a

+5)=(

a

+5)(4

x

-4) =4(

a

+5)(

x

-1) ( 3 )3

x

(

x

-

y

)-9(

x

-

y

)=(

x

-

y

)(3

x

-9) =3(

x

-

y

)(

x

-3) ( 4 )2

x

(

a

+

b

)+6

y

(

a

+

b

)=(

a

+

b

)(2

x

+6

y

) =2(

a

+

b

)(

x

+3

y

) ( 5 )6

a

(

x

+

y

)-8

b

(

x

+

y

)=(

x

+

y

)(6

a

-8

b

) =2(

x

+

y

)(3

a

-4

b

) ( 6 )6

a

(

a

-

b

)+9

b

(

a

-

b

)=(

a

-

b

)(6

a

+9

b

) =3(

a

-

b

)(2

a

+3

b

) ( 7 )3

ab

(

x

+

y

)+3

bc

(

x

+

y

)=(

x

+

y

)(3

ab

+3

bc

) =3

b

(

x

+

y

)(

a

+

c

) ( 8 )6

ax

(

x

-

y

)+6

bx

(

x

-

y

)=(

x

-

y

)(6

ax

+6

bx

) =6

x

(

x

-

y

)(

a

+

b

) ( 9 )3

x

2(

x

+4)-

xy

(

x

+4)=(

x

+4)(3

x

2-

xy

) =

x

(

x

+4)(3

x

-

y

) (10)5

a

2(

x

-5)-

a

(

x

-5)=(

x

-5)(5

a

2-

a

) =

a

(

x

-5)(5

a

-1) (11)8

x

2(

a

+

b

)+2

x

(

a

+

b

)=(

a

+

b

)(8

x

2+2

x

) =2

x

(

a

+

b

)(4

x

+1) (12)

x

2

y

(

a

-1)+

xy

2(

a

-1)=(

a

-1)(

x

2

y

+

xy

2) =

xy

(

a

-1)(

x

+

y

) (13)4

axy

(

x

+

y

)-6

bxy

(

x

+

y

) =(

x

+

y

)(4

axy

-6

bxy

) =2

xy

(

x

+

y

)(2

a

-3

b

) (14)9

ax

2(

x

-

y

)-6

bx

2(

x

-

y

) =(

x

-

y

)(9

ax

2-6

bx

2) =3

x

2(

x

-

y

)(3

a

-2

b

) M - 67 ( 1 )

x

(

a

+

b

) ( 2 )5(

a

-

b

) ( 3 )-4

y

(

x

-2

z

) ( 4 )3

x

(2

x

-3

y

) ( 5 )

xy

(

x

+

y

) ( 6 )-

ax

2(7

a

+4) ( 7 )3

a

2

b

2(2

b

+

a

) ( 8 )2

xyz

2(4

xz

-3

y

) ( 9 )

x

(

a

+

b

+

c

) (10)

ab

(

a

-

b

+1) (11)4

a

(2

x

2 -

y

2-3

z

2) (12)(

a

+

b

)(

x

+

y

) (13)(2

x

-

y

)(1+5

x

) (14)7(

a

+3)(

x

-1) (15)2

x

(

x

-4)(2

x

-

y

) (16)3

x

2(

x

-

y

)(4

a

+3

b

) <풀이> ( 1 )

ax

+

bx

=

x

×

a

+

x

×

b

=

x

(

a

+

b

) ( 2 )5

a

-5

b

=5×

a

-5×

b

=5(

a

-

b

) ( 3 )-4

xy

+8

yz

=(-4

y

x

-(-4

y

)×2

z

=-4

y

(

x

-2

z

) ( 4 )6

x

2-9

xy

=3

x

×2

x

-3

x

×3

y

=3

x

(2

x

-3

y

) ( 5 )

x

2

y

+

xy

2=

xy

×

x

+

xy

×

y

=

xy

(

x

+

y

) ( 6 )-7

a

2

x

2-4

ax

2=(-

ax

2)×7

a

+(-

ax

2)×4 =-

ax

2(7

a

+4) ( 7 )6

a

2

b

3+3

a

3

b

2=3

a

2

b

2 ×2

b

+3

a

2

b

2 ×

a

=3

a

2

b

2(2

b

+

a

) ( 8 )8

x

2

yz

3-6

xy

2

z

2=2

xyz

2 ×4

xz

-2

xyz

2 ×3

y

=2

xyz

2(4

xz

-3

y

) ( 9 )

ax

+

bx

+

cx

=

x

×

a

+

x

×

b

+

x

×

c

=

x

(

a

+

b

+

c

)

(5)

M - 68 ( 1 )

x

, 1 ( 2 )3, 3, 3 ( 3 )(

x

+5)2 ( 4 )(

x

-2)2 ( 5 ) (

x

+4)2 ( 6 )(

x

-6)2 ( 7 )(

x

+7)2 ( 8 )(

x

-8)2 ( 9 )(

x

+9)2 (10)(

x

-1)2 (11)(

x

+3)2 (12)(

x

-4)2 (13)(

x

+6)2 (14)(

x

-5)2 <풀이> ※ 주어진 식을

a

2+2

ab

+

b

2,

a

2-2

ab

+

b

2의 꼴 로 고친다. ( 1 )

x

2+2

x

+1=

x

2+2×

x

×1+12=(

x

+1)2 ( 2 )

x

2-6

x

+9=

x

2-2×

x

×3+32=(

x

-3)2 ( 3 )

x

2+10

x

+25=

x

2+2×

x

×5+52=(

x

+5)2 ( 4 )

x

2-4

x

+4=

x

2-2×

x

×2+22=(

x

-2)2 ( 5 )

x

2+8

x

+16=

x

2+2×

x

×4+42=(

x

+4)2 ( 6 )

x

2-12

x

+36=

x

2-2×

x

×6+62=(

x

-6)2 ( 7 )

x

2+14

x

+49=

x

2+2×

x

×7+72=(

x

+7)2 ( 8 )

x

2-16

x

+64=

x

2-2×

x

×8+82=(

x

-8)2 ( 9 )

x

2+18

x

+81=

x

2+2×

x

×9+92=(

x

+9)2 (10)

x

2-2

x

+1=

x

2-2×

x

×1+12=(

x

-1)2 (11)

x

2+6

x

+9=

x

2+2×

x

×3+32=(

x

+3)2 (12)

x

2-8

x

+16=

x

2-2×

x

×4+42=(

x

-4)2 (13)

x

2+12

x

+36=

x

2+2×

x

×6+62=(

x

+6)2 (14)

x

2-10

x

+25=

x

2-2×

x

×5+52=(

x

-5)2 ●2+2×●×▲+▲2 (● + ▲)2 � � ●2-2×●×▲+▲2 (● - ▲)2 � � M - 69 ( 1 )2

x

, 1 ( 2 )2, 2, 2 ( 3 )(2

x

+5)2 ( 4 )(3

x

-5)2 ( 5 ) (3

x

+1)2 ( 6 )(2

x

-3)2 ( 7 )(3

x

+4)2 ( 8 )(2

x

-7)2 ( 9 )(2

x

+9)2 (10)(3

x

-7)2 (11)(4

x

+3)2 (12)(6

x

-5)2 (13)(3

x

+8)2 (14)(5

x

-4)2 (15)(7

x

+2)2 (16)(3

x

-10)2 <풀이> ( 1 )4

x

2+4

x

+1=(2

x

)2+2×2

x

×1+12=(2

x

+1)2 ( 2 )9

x

2-12

x

+4=(3

x

)2 -2×3

x

×2+22=(3

x

-2)2 ( 3 )4

x

2+20

x

+25=(2

x

)2+2×2

x

×5+52 =(2

x

+5)2 ( 4 )9

x

2-30

x

+25=(3

x

)2-2×3

x

×5+52 =(3

x

-5)2 ( 5 )9

x

2+6

x

+1=(3

x

)2+2×3

x

×1+12=(3

x

+1)2 ( 6 )4

x

2-12

x

+9=(2

x

)2-2×2

x

×3+32=(2

x

-3)2 ( 7 )9

x

2+24

x

+16=(3

x

)2+2×3

x

×4+42 =(3

x

+4)2 ( 8 )4

x

2-28

x

+49=(2

x

)2-2×2

x

×7+72 =(2

x

-7)2 ( 9 )4

x

2+36

x

+81=(2

x

)2+2×2

x

×9+92 =(2

x

+9)2 (10)9

x

2-42

x

+49=(3

x

)2-2×3

x

×7+72 =(3

x

-7)2 (11)16

x

2+24

x

+9=(4

x

)2+2×4

x

×3+32 =(4

x

+3)2 (12)36

x

2-60

x

+25=(6

x

)2-2×6

x

×5+52 =(6

x

-5)2 (13)9

x

2+48

x

+64=(3

x

)2+2×3

x

×8+82 =(3

x

+8)2 (14)25

x

2-40

x

+16=(5

x

)2-2×5

x

×4+42 =(5

x

-4)2 (15)49

x

2+28

x

+4=(7

x

)2+2×7

x

×2+22 =(7

x

+2)2 (16)9

x

2-60

x

+100=(3

x

)2-2×3

x

×10+102 =(3

x

-10)2 M - 70 ( 1 )(

x

+

y

)2 ( 2 )

x

,

y

( 3 )4

y

, 4

y

, 4

y

( 4 )(

x

-7

y

)2 ( 5 ) (

x

+2

y

)2 ( 6 )(

x

-3

y

)2 ( 7 )(

x

+5

y

)2 ( 8 )(

x

-9

y

)2 ( 9 )(

x

+3

y

)2 (10)(

x

-2

y

)2 (11)(

x

+6

y

)2 (12)(

x

-5

y

)2 (13)(

x

+7

y

)2 (14)(

x

-4

y

)2 (15)(

x

+9

y

)2 (16)(

x

-8

y

)2 (10)

a

2

b

-

ab

2+

ab

=

ab

×

a

-

ab

×

b

+

ab

×1 =

ab

(

a

-

b

+1) (11)8

ax

2-4

ay

2-12

az

2 =4

a

×2

x

2-4

a

×

y

2-4

a

×3

z

2 =4

a

(2

x

2-

y

2-3

z

2) (12)(

a

+

b

)

x

+(

a

+

b

)

y

=(

a

+

b

)(

x

+

y

) (13)(2

x

-

y

)+5

x

(2

x

-

y

)=(2

x

-

y

)(1+5

x

) (14)7

x

(

a

+3)-7(

a

+3) =(

a

+3)(7

x

-7) =7(

a

+3)(

x

-1) (15)4

x

2(

x

-4)-2

xy

(

x

-4) =(

x

-4)(4

x

2 -2

xy

) =2

x

(

x

-4)(2

x

-

y

) (16)12

ax

2(

x

-

y

)+9

bx

2(

x

-

y

) =(

x

-

y

)(12

ax

2+9

bx

2) =3

x

2(

x

-

y

)(4

a

+3

b

)

(6)

<풀이> ( 1 )

x

2+2

xy

+

y

2=

x

2+2×

x

×

y

+

y

2=(

x

+

y

)2 ( 2 )

x

2-2

xy

+

y

2=

x

2-2×

x

×

y

+

y

2=(

x

-

y

)2 ( 3 )

x

2+8

xy

+16

y

2=

x

2+2×

x

×4

y

+(4

y

)2 =(

x

+4

y

)2 ( 4 )

x

2-14

xy

+49

y

2=

x

2-2×

x

×7

y

+(7

y

)2 =(

x

-7

y

)2 ( 5 )

x

2+4

xy

+4

y

2=

x

2+2×

x

×2

y

+(2

y

)2 =(

x

+2

y

)2 ( 6 )

x

2-6

xy

+9

y

2=

x

2-2×

x

×3

y

+(3

y

)2 =(

x

-3

y

)2 ( 7 )

x

2+10

xy

+25

y

2=

x

2+2×

x

×5

y

+(5

y

)2 =(

x

+5

y

)2 ( 8 )

x

2-18

xy

+81

y

2=

x

2-2×

x

×9

y

+(9

y

)2 =(

x

-9

y

)2 ( 9 )

x

2+6

xy

+9

y

2=

x

2+2×

x

×3

y

+(3

y

)2 =(

x

+3

y

)2 (10)

x

2-4

xy

+4

y

2=

x

2-2×

x

×2

y

+(2

y

)2 =(

x

-2

y

)2 (11)

x

2+12

xy

+36

y

2=

x

2+2×

x

×6

y

+(6

y

)2 =(

x

+6

y

)2 (12)

x

2-10

xy

+25

y

2=

x

2-2×

x

×5

y

+(5

y

)2 =(

x

-5

y

)2 (13)

x

2+14

xy

+49

y

2=

x

2+2×

x

×7

y

+(7

y

)2 =(

x

+7

y

)2 (14)

x

2-8

xy

+16

y

2=

x

2-2×

x

×4

y

+(4

y

)2 =(

x

-4

y

)2 (15)

x

2+18

xy

+81

y

2=

x

2+2×

x

×9

y

+(9

y

)2 =(

x

+9

y

)2 (16)

x

2-16

xy

+64

y

2=

x

2-2×

x

×8

y

+(8

y

)2 =(

x

-8

y

)2 M - 71 ( 1 )2

x

, 2

x

, 2

x

( 2 )(3

x

-4

y

)2 ( 3 )(2

x

+5

y

)2 ( 4 )7

y

, 7

y

, 7

y

( 5 ) (3

x

+2

y

)2 ( 6 )(2

x

-7

y

)2 ( 7 )(3

x

+5

y

)2 ( 8 )(2

x

-9

y

)2 ( 9 )(2

x

+3

y

)2 (10)(3

x

-

y

)2 (11)(4

x

+7

y

)2 (12)(5

x

-8

y

)2 (13)(3

x

+8

y

)2 (14)(7

x

-5

y

)2 (15)(3

x

+10

y

)2 (16)(6

x

-7

y

)2 <풀이> ( 1 )4

x

2+4

xy

+

y

2=(2

x

)2+2×2

x

×

y

+

y

2 =(2

x

+

y

)2 ( 2 )9

x

2-24

xy

+16

y

2=(3

x

)2 -2×3

x

×4

y

+(4

y

)2 =(3

x

-4

y

)2 ( 3 )4

x

2+20

xy

+25

y

2=(2

x

)2+2×2

x

×5

y

+(5

y

)2 =(2

x

+5

y

)2 ( 4 )9

x

2-42

xy

+49

y

2=(3

x

)2-2×3

x

×7

y

+(7

y

)2 =(3

x

-7

y

)2 ( 5 )9

x

2+12

xy

+4

y

2=(3

x

)2+2×3

x

×2

y

+(2

y

)2 =(3

x

+2

y

)2 ( 6 )4

x

2-28

xy

+49

y

2=(2

x

)2-2×2

x

×7

y

+(7

y

)2 =(2

x

-7

y

)2 ( 7 )9

x

2+30

xy

+25

y

2=(3

x

)2+2×3

x

×5

y

+(5

y

)2 =(3

x

+5

y

)2 ( 8 )4

x

2-36

xy

+81

y

2=(2

x

)2-2×2

x

×9

y

+(9

y

)2 =(2

x

-9

y

)2 ( 9 )4

x

2+12

xy

+9

y

2=(2

x

)2+2×2

x

×3

y

+(3

y

)2 =(2

x

+3

y

)2 (10)9

x

2-6

xy

+

y

2=(3

x

)2-2×3

x

×

y

+

y

2 =(3

x

-

y

)2 (11)16

x

2+56

xy

+49

y

2=(4

x

)2+2×4

x

×7

y

+(7

y

)2 =(4

x

+7

y

)2 (12)25

x

2-80

xy

+64

y

2=(5

x

)2-2×5

x

×8

y

+(8

y

)2 =(5

x

-8

y

)2 (13)9

x

2+48

xy

+64

y

2=(3

x

)2+2×3

x

×8

y

+(8

y

)2 =(3

x

+8

y

)2 (14)49

x

2-70

xy

+25

y

2=(7

x

)2-2×7

x

×5

y

+(5

y

)2 =(7

x

-5

y

)2 (15)9

x

2+60

xy

+100

y

2 =(3

x

)2+2×3

x

×10

y

+(10

y

)2 =(3

x

+10

y

)2 (16)36

x

2-84

xy

+49

y

2=(6

x

)2 -2×6

x

×7

y

+(7

y

)2 =(6

x

-7

y

)2 M - 72 ( 1 )

xy

, 2,

xy

( 2 )(

ax

-3)2 ( 3 )(2

xy

+5)2 ( 4 )(3

ab

-7

x

)2 ( 5 ) , , ( 6 )&

x

- *2

( 7 )&

x

+ *2 ( 8 )&

x

- *2 ( 9 )&2

x

+ *2

(10)&3

x

- *2 (11)&2

x

+ *2 (12)&3

x

- *2

(13)&3

x

+

y

*2 (14)&2

x

-

y

*2 (15)&3

x

+

y

*2

(16)&4

x

-34

y

*2 2 3 1 4 1 2 2 3 3 4 1 3 1 2 3 4 2 3 1 3 1 2 1 2 1 2 <풀이> ( 1 )

x

2

y

2+4

xy

+4=(

xy

)2+2×

xy

×2+22 =(

xy

+2)2

(7)

( 2 )

a

2

x

2-6

ax

+9=(

ax

)2-2×

ax

×3+32 =(

ax

-3)2 ( 3 )4

x

2

y

2+20

xy

+25=(2

xy

)2+2×2

xy

×5+52 =(2

xy

+5)2 ( 4 )9

a

2

b

2-42

abx

+49

x

2 =(3

ab

)2-2×3

ab

×7

x

+(7

x

)2 =(3

ab

-7

x

)2 ( 5 )

x

2+

x

+ =

x

2+2×

x

× +& * 2 ( 5 )

x

2+

x

+ =&

x

+ *2 ( 6 )

x

2-

x

+ =

x

2-2×

x

× +& * 2 ( 5 )

x

2-

x

+ =&

x

- *2 ( 7 )

x

2+

x

+ =

x

2+2×

x

× +& * 2 ( 7 )

x

2+

x

+ =&

x

+ *2 ( 8 )

x

2-

x

+ =

x

2-2×

x

× +& * 2 ( 7 )

x

2+

x

+ =&

x

- *2 ( 9 )4

x

2+2

x

+ =(2

x

)2 +2×2

x

× +& * 2 ( 9 )4

x

2+2

x

+ =&2

x

+ *2 (10)9

x

2-2

x

+ =(3

x

)2-2×3

x

× +& * 2 ( 9 )4

x

2+2

x

+ =&3

x

- *2 (11)4

x

2+3

x

+ =(2

x

)2 +2×2

x

× +& * 2 ( 9 )4

x

2+2

x

+ = &2

x

+ *2 (12)9

x

2-4

x

+ =(3

x

)2-2×3

x

× +& * 2 ( 9 )4

x

2+2

x

+ =&3

x

- *2 (13)9

x

2+3

xy

+

y

2=(3

x

)2+2×3

x

×

y

+&

y

*2 ( 9 )4

x

2+2

xy

+

y

2=&3

x

+

y

*2 (14)4

x

2-

xy

+

y

2=(2

x

)2-2×2

x

×

y

+&

y

*2 ( 9 )4

x

2+

xy

+

y

2=&2

x

-1

y

*2 4 1 4 1 4 1 4 1 16 1 2 1 4 1 2 1 2 1 4 2 3 1 4 2 3 2 3 4 9 3 4 1 4 3 4 3 4 9 16 1 3 1 4 1 3 1 3 1 9 1 2 1 4 1 2 1 2 1 4 3 4 1 4 3 4 3 4 9 16 3 2 2 3 1 4 1 4 2 3 2 3 4 9 4 3 1 3 1 3 1 3 1 9 2 3 1 2 1 4 1 2 1 2 1 4 (15)9

x

2+4

xy

+

y

2=(3

x

)2+2×3

x

×

y

+&

y

*2 ( 9 )4

x

2+2

xy

+

y

2=&3

x

+

y

*2 (16)16

x

2-6

xy

+

y

2 ( 9 )=(4

x

)2 -2×4

x

×

y

+&

y

*2 ( 9 )=&4

x

-34

y

*2 3 4 3 4 9 16 2 3 1 4 2 3 2 3 4 9 M - 73 ( 1 )5, 5, 1 ( 2 )2(

x

-3)2 ( 3 )3(3

x

+2)2 ( 4 )4, 4, 2

y

( 5 ) 3(

x

+3

y

)2 ( 6 )5(2

x

-3

y

)2 ( 7 )

x

(

x

+4)2 ( 8 )2

x

(3

x

-1)2 ( 9 )3

x

(

a

+3

b

)2 (10)2

a

(2

xy

-3)2 (11) 3

ab

(3

x

+2

y

)2 (12)-(

x

-

y

)2 (13)-(2

x

+5

y

)2 (14)-(

x

2+

y

2)2 <풀이> ( 1 )5

x

2+10

x

+5=5(

x

2+2

x

+1

)

=5(

x

2+2×

x

×1+12) =5(

x

+1)2 ( 2 )2

x

2-12

x

+18=2(

x

2 -6

x

+9) =2(

x

2-2×

x

×3+32) =2(

x

-3)2 ( 3 )27

x

2+36

x

+12=3(9

x

2+12

x

+4) =3{(3

x

)2 +2×3

x

×2+22} =3(3

x

+2)2 ( 4 )4

x

2-16

xy

+16

y

2=4(

x

2-4

xy

+4

y

2) =4{

x

2-2×

x

×2

y

+(2

y

)2} =4(

x

-2

y

)2 ( 5 )3

x

2+18

xy

+27

y

2=3(

x

2+6

xy

+9

y

2) =3{

x

2+2×

x

×3

y

+(3

y

)2} =3(

x

+3

y

)2 ( 6 )20

x

2-60

xy

+45

y

2 =5(4

x

2-12

xy

+9

y

2) =5{(2

x

)2-2×2

x

×3

y

+(3

y

)2} =5(2

x

-3

y

)2 ( 7 )

x

3+8

x

2+16

x

=

x

(

x

2 +8

x

+16) =

x

(

x

2+2×

x

×4+42) =

x

(

x

+4)2 ( 8 )18

x

3-12

x

2+2

x

=2

x

(9

x

2-6

x

+1) =2

x

{(3

x

)2 -2×3

x

×1+12} =2

x

(3

x

-1)2

(8)

( 9 )3

a

2

x

+18

abx

+27

b

2

x

=3

x

(

a

2+6

ab

+9

b

2) =3

x

{

a

2+2×

a

×3

b

+(3

b

)2} =3

x

(

a

+3

b

)2 (10)8

ax

2

y

2-24

axy

+18

a

=2

a

(4

x

2

y

2-12

xy

+9) =2

a

{(2

xy

)2-2×2

xy

×3+32} =2

a

(2

xy

-3)2

(11)27

abx

2+36

abxy

+12

aby

2

=3

ab

(9

x

2+12

xy

+4

y

2) =3

ab

{(3

x

)2+2×3

x

×2

y

+(2

y

)2} =3

ab

(3

x

+2

y

)2 (12)-

x

2+2

xy

-

y

2=-(

x

2-2

xy

+

y

2) =-(

x

2-2×

x

×

y

+

y

2) =-(

x

-

y

)2 (13)-4

x

2-20

xy

-25

y

2 =-(4

x

2+20

xy

+25

y

2) =-{(2

x

)2+2×2

x

×5

y

+(5

y

)2} =-(2

x

+5

y

)2 (14)-

x

4-2

x

2

y

2-

y

4=-(

x

4 +2

x

2

y

2+

y

4) =-{(

x

2)2+2×

x

2 ×

y

2+(

y

2)2} =-(

x

2+

y

2)2 M - 74 ( 1 )(

x

+2)2 ( 2 )(

x

-7)2 ( 3 )(

x

+8)2 ( 4 )(2

x

-5)2 ( 5 ) (3

x

+7)2 ( 6 )(5

x

-4)2 ( 7 )(

x

+6

y

)2 ( 8 )(

x

-9

y

)2 ( 9 )(3

x

+4

y

)2 (10)(4

x

-5

y

)2 (11)(

xy

+3)2 (12) &

x

- *2 (13)&2

x

+ *2 (14)5(

x

-2)2 (15)3(

x

+4

y

)2 (16)-

x

(

x

-5)2 3 2 1 2 <풀이> ( 1 )

x

2+4

x

+4=

x

2+2×

x

×2+22=(

x

+2)2 ( 2 )

x

2-14

x

+49=

x

2-2×

x

×7+72=(

x

-7)2 ( 3 )

x

2+16

x

+64=

x

2+2×

x

×8+82=(

x

+8)2 ( 4 )4

x

2-20

x

+25=(2

x

)2-2×2

x

×5+52 =(2

x

-5)2 ( 5 )9

x

2+42

x

+49=(3

x

)2 +2×3

x

×7+72 =(3

x

+7)2 ( 6 )25

x

2-40

x

+16=(5

x

)2-2×5

x

×4+42 =(5

x

-4)2 ( 7 )

x

2+12

xy

+36

y

2=

x

2+2×

x

×6

y

+(6

y

)2 =(

x

+6

y

)2 ( 8 )

x

2-18

xy

+81

y

2=

x

2-2×

x

×9

y

+(9

y

)2 =(

x

-9

y

)2 ( 9 )9

x

2+24

xy

+16

y

2=(3

x

)2+2×3

x

×4

y

+(4

y

)2 =(3

x

+4

y

)2 (10)16

x

2-40

xy

+25

y

2=(4

x

)2-2×4

x

×5

y

+(5

y

)2 =(4

x

-5

y

)2 (11)

x

2

y

2+6

xy

+9=(

xy

)2+2×

xy

×3+32=(

xy

+3)2 (12)

x

2-

x

+ =

x

2-2×

x

× +& * 2 =&

x

- *2 (13)4

x

2+6

x

+ =(2

x

)2+2×2

x

× +& * 2 (13)4

x

2+6

x

+ =&2

x

+ *2 (14)5

x

2-20

x

+20=5(

x

2-4

x

+4) =5(

x

2-2×

x

×2+22) =5(

x

-2)2 (15)3

x

2+24

xy

+48

y

2=3(

x

2 +8

xy

+16

y

2) =3{

x

2+2×

x

×4

y

+(4

y

)2} =3(

x

+4

y

)2 (16)-

x

3+10

x

2-25

x

=-

x

(

x

2-10

x

+25) =-

x

(

x

2 -2×

x

×5+52) =-

x

(

x

-5)2 3 2 9 4 3 2 3 2 9 4 1 2 1 2 1 2 1 4 M - 75 ( 1 )

y

(

x

+

z

) ( 2 )4(2

a

-3

b

) ( 3 )

ab

(

x

+5) ( 4 )3

xz

(3

xy

-2

z

) ( 5 ) 3

x

(

x

+

y

+3) ( 6 )2

abc

(4

a

-3

b

+5) ( 7 )(

x

-6)(1+

a

) ( 8 )5

x

(

a

+2)(

x

-2) ( 9 )(

x

+4)2 (10)(2

x

-7)2 (11)(3

x

+5)2 (12)(

x

-8

y

)2 (13)(4

x

+9

y

)2 (14)&3

x

- *2 (15)3(2

x

-3)2 (16) 2

a

(

x

+4

y

)2 4 3 <풀이> ( 1 )

xy

+

yz

=

y

×

x

+

y

×

z

=

y

(

x

+

z

) ( 2 )8

a

-12

b

=4×2

a

-4×3

b

=4(2

a

-3

b

) ( 3 )

abx

+5

ab

=

ab

×

x

+

ab

×5=

ab

(

x

+5) ( 4 )9

x

2

yz

-6

xz

2=3

xz

×3

xy

-3

xz

×2

z

=3

xz

(3

xy

-2

z

) ( 5 )3

x

2+3

xy

+9

x

=3

x

×

x

+3

x

×

y

+3

x

×3 =3

x

(

x

+

y

+3) ( 6 )8

a

2

bc

-6

ab

2

c

+10

abc

=2

abc

×4

a

-2

abc

×3

b

+2

abc

×5 =2

abc

(4

a

-3

b

+5)

( 7 )(

x

-6)+

a

(

x

-6)=(

x

-6)(1+

a

)

( 8 )5

x

2(

a

+2)-10

x

(

a

+2)=(

a

+2)(5

x

2-10

x

)

(9)

( 9 )

x

2+8

x

+16=

x

2+2×

x

×4+42=(

x

+4)2 (10)4

x

2-28

x

+49=(2

x

)2-2×2

x

×7+72=(2

x

-7)2 (11)9

x

2+30

x

+25=(3

x

)2+2×3

x

×5+52 =(3

x

+5)2 (12)

x

2-16

xy

+64

y

2=

x

2-2×

x

×8

y

+(8

y

)2 =(

x

-8

y

)2 (13)16

x

2+72

xy

+81

y

2=(4

x

)2+2×4

x

×9

y

+(9

y

)2 =(4

x

+9

y

)2 (14)9

x

2-8

x

+ =(3

x

)2-2×3

x

× +& * 2 (13)4

x

2+6

x

+ =&3

x

- *2 (15)12

x

2-36

x

+27=3(4

x

2-12

x

+9) =3{(2

x

)2 -2×2

x

×3+32} =3(2

x

-3)2 (16)2

ax

2+16

axy

+32

ay

2 =2

a

(

x

2+8

xy

+16

y

2) =2

a

{

x

2 +2×

x

×4

y

+(4

y

)2} =2

a

(

x

+4

y

)2 4 3 9 4 4 3 4 3 16 9 M - 76 ( 1 )1, 1 ( 2 )(

x

+2)(

x

-2) ( 3 )(

x

+3)(

x

-3) ( 4 )(

x

+4)(

x

-4) ( 5 ) 6, 6, 6 ( 6 )(

x

+7)(

x

-7) ( 7 )(

x

+8)(

x

-8) ( 8 )(

x

+9)(

x

-9) ( 9 )2, 2 (10)(3+

x

)(3-

x

) (11)(6+

x

)(6-

x

) (12)(8+

x

)(8-

x

) (13)(9+

x

)(9-

x

) (14)1, 1, 1 (15)(4+

x

)(4-

x

) (16)(5+

x

)(5-

x

) (17)(7+

x

)(7-

x

) (18)(10+

x

)(10-

x

) <풀이> ( 1 )

x

2-1=

x

2-12=(

x

+1)(

x

-1) ( 2 )

x

2-4=

x

2-22=(

x

+2)(

x

-2) ( 3 )

x

2-9=

x

2-32=(

x

+3)(

x

-3) ( 4 )

x

2-16=

x

2-42=(

x

+4)(

x

-4) ( 5 )

x

2-36=

x

2-62=(

x

+6)(

x

-6) ( 6 )

x

2-49=

x

2-72=(

x

+7)(

x

-7) ( 7 )

x

2-64=

x

2-82=(

x

+8)(

x

-8) ( 8 )

x

2-81=

x

2-92=(

x

+9)(

x

-9) ( 9 )4-

x

2=22-

x

2=(2+

x

)(2-

x

) (10)9-

x

2=32-

x

2=(3+

x

)(3-

x

) (11)36-

x

2=62-

x

2=(6+

x

)(6-

x

) (12)64-

x

2=82-

x

2=(8+

x

)(8-

x

) (13)81-

x

2=92-

x

2=(9+

x

)(9-

x

) (14)-

x

2+1=1-

x

2=12-

x

2=(1+

x

)(1-

x

) <다른 풀이> -

x

2+1=-(

x

2-1)=-(

x

2-12)=-(

x

+1)(

x

-1) (15)-

x

2+16=16-

x

2=42-

x

2=(4+

x

)(4-

x

) (16)-

x

2+25=25-

x

2=52-

x

2=(5+

x

)(5-

x

) (17)-

x

2+49=49-

x

2=72-

x

2=(7+

x

)(7-

x

) (18)-

x

2+100=100-

x

2=102-

x

2=(10+

x

)(10-

x

) M - 77 ( 1 )(

x

+

y

)(

x

-

y

) ( 2 )4

y

, 4

y

( 3 )(

x

+5

y

)(

x

-5

y

) ( 4 )(

x

+8

y

)(

x

-8

y

) ( 5 ) (

x

+9

y

)(

x

-9

y

) ( 6 )(

x

+2

y

)(

x

-2

y

) ( 7 )(

x

+3

y

)(

x

-3

y

) ( 8 )(

x

+6

y

)(

x

-6

y

) ( 9 )(

x

+7

y

)(

x

-7

y

) (10)(

x

+10

y

)(

x

-10

y

) (11) , , (12)&

x

+ *&

x

- *

(13)& +

x

*& -

x

* (14)& +

x

*& -

x

* (15)&

x

+

y

*&

x

-

y

* (16)&

x

+

y

*&

x

-

y

*

(17)& +

x

*& -

x

* (18)& +

x

*& -

x

* (19)(

xy

+9)(

xy

-9) (20)(8+

xy

)(8-

xy

) 3 4 3 4 2 3 2 3 1 6 1 6 1 3 1 3 1 7 1 7 1 5 1 5 1 4 1 4 1 2 1 2 1 2 <풀이> ( 1 )

x

2-

y

2=(

x

+

y

)(

x

-

y

) ( 2 )

x

2-16

y

2=

x

2-(4

y

)2=(

x

+4

y

)(

x

-4

y

) ( 3 )

x

2-25

y

2=

x

2-(5

y

)2=(

x

+5

y

)(

x

-5

y

) ( 4 )

x

2-64

y

2=

x

2-(8

y

)2=(

x

+8

y

)(

x

-8

y

) ( 5 )

x

2-81

y

2=

x

2-(9

y

)2=(

x

+9

y

)(

x

-9

y

) ( 6 )

x

2-4

y

2=

x

2-(2

y

)2=(

x

+2

y

)(

x

-2

y

) ( 7 )

x

2-9

y

2=

x

2-(3

y

)2=(

x

+3

y

)(

x

-3

y

) ( 8 )

x

2-36

y

2=

x

2-(6

y

)2=(

x

+6

y

)(

x

-6

y

) ( 9 )

x

2-49

y

2=

x

2-(7

y

)2=(

x

+7

y

)(

x

-7

y

) (10)

x

2-100

y

2=

x

2-(10

y

)2=(

x

+10

y

)(

x

-10

y

)

(11)

x

2- =

x

2-& *2=&

x

+ *&

x

- *

(12)

x

2- =

x

2-& *2=&

x

+ *&

x

- *

(13) -

x

2=& *2-

x

2=& +

x

*& -

x

*

(14) -

x

2=& *2-

x

2=& +

x

*& -

x

*

(15)

x

2-

y

2=

x

2

-&

y

*2=&

x

+

y

*&

x

-

y

*

(16)

x

2-

y

2=

x

2-&

y

*2=&

x

+

y

*&

x

-1

y

*

6 1 6 1 6 1 36 1 3 1 3 1 3 1 9 1 7 1 7 1 7 1 49 1 5 1 5 1 5 1 25 1 4 1 4 1 4 1 16 1 2 1 2 1 2 1 4

(10)

(17)-

x

2+ = -

x

2=& *2-

x

2=& +

x

*& -

x

* (18)-

x

2+ = -

x

2=& *2-

x

2 (18)-

x

2+ = -

x

2=& +

x

*& -

x

* (19)

x

2

y

2-81=(

xy

)2-92=(

xy

+9)(

xy

-9) (20)64-

x

2

y

2=82-(

xy

)2=(8+

xy

)(8-

xy

) 3 4 3 4 9 16 9 16 3 4 9 16 9 16 2 3 2 3 2 3 4 9 4 9 M - 78 ( 1 )2

x

, 2

x

( 2 )(3

x

+2)(3

x

-2) ( 3 )(2

x

+5)(2

x

-5) ( 4 )(3

x

+7)(3

x

-7) ( 5 ) (2

x

+9)(2

x

-9) ( 6 )(3

x

+8)(3

x

-8) ( 7 )(6

x

+1)(6

x

-1) ( 8 )(4

x

+3)(4

x

-3) ( 9 )(5

x

+6)(5

x

-6) (10)(7

x

+4)(7

x

-4) (11)(3+2

x

)(3-2

x

) (12)(1+3

x

)(1-3

x

) (13)(6+2

x

)(6-2

x

) (14)3

x

, 3

x

, 3

x

(15)(7+2

x

)(7-2

x

) (16)(2+3

x

)(2-3

x

) (17)(3+4

x

)(3-4

x

) (18)(8+5

x

)(8-5

x

) (19)(4+9

x

)(4-9

x

) (20)(9+8

x

)(9-8

x

) <풀이> ( 1 )4

x

2-1=(2

x

)2-12=(2

x

+1)(2

x

-1) ( 2 )9

x

2-4=(3

x

)2 -22=(3

x

+2)(3

x

-2) ( 3 )4

x

2-25=(2

x

)2 -52=(2

x

+5)(2

x

-5) ( 4 )9

x

2-49=(3

x

)2-72=(3

x

+7)(3

x

-7) ( 5 )4

x

2-81=(2

x

)2-92=(2

x

+9)(2

x

-9) ( 6 )9

x

2-64=(3

x

)2 -82=(3

x

+8)(3

x

-8) ( 7 )36

x

2-1=(6

x

)2 -12=(6

x

+1)(6

x

-1) ( 8 )16

x

2-9=(4

x

)2-32=(4

x

+3)(4

x

-3) ( 9 )25

x

2-36=(5

x

)2-62=(5

x

+6)(5

x

-6) (10)49

x

2-16=(7

x

)2 -42=(7

x

+4)(7

x

-4) (11)9-4

x

2=32-(2

x

)2=(3+2

x

)(3-2

x

) (12)1-9

x

2=12-(3

x

)2=(1+3

x

)(1-3

x

) (13)36-4

x

2=62-(2

x

)2=(6+2

x

)(6-2

x

) (14)25-9

x

2=52-(3

x

)2=(5+3

x

)(5-3

x

) (15)49-4

x

2=72-(2

x

)2=(7+2

x

)(7-2

x

) (16)-9

x

2+4=4-9

x

2=22-(3

x

)2=(2+3

x

)(2-3

x

) (17)-16

x

2+9=9-16

x

2=32-(4

x

)2 =(3+4

x

)(3-4

x

) (18)-25

x

2+64=64-25

x

2=82-(5

x

)2 =(8+5

x

)(8-5

x

) (19)-81

x

2+16=16-81

x

2=42-(9

x

)2 =(4+9

x

)(4-9

x

) (20)-64

x

2+81=81-64

x

2=92-(8

x

)2 =(9+8

x

)(9-8

x

) M - 79 ( 1 )(2

x

+

y

)(2

x

-

y

) ( 2 )(3

x

+4

y

)(3

x

-4

y

) ( 3 )(2

x

+7

y

)(2

x

-7

y

) ( 4 )(3

x

+8

y

)(3

x

-8

y

) ( 5 ) (2

x

+9

y

)(2

x

-9

y

) ( 6 )(3

x

+2

y

)(3

x

-2

y

) ( 7 )(4

x

+5

y

)(4

x

-5

y

) ( 8 )(8

x

+

y

)(8

x

-

y

) ( 9 )(7

x

+3

y

)(7

x

-3

y

) (10)(5

x

+6

y

)(5

x

-6

y

)

(11)&2

x

+ *&2

x

- * (12)&3

x

+ *&3

x

- *

(13)&5+

x

*&5-

x

* (14)&4+

x

*&4-

x

* (15)&2

x

+

y

*&2

x

-

y

*(16)&3

x

+

y

*&3

x

-

y

*

(17)& +4

x

*& -4

x

* (18)& +

x

*& -

x

* (19)(5

xy

+7)(5

xy

-7) (20)(9+6

xy

)(9-6

xy

) 1 3 1 2 1 3 1 2 2 3 2 3 3 4 3 4 2 3 2 3 1 5 1 5 1 4 1 4 1 2 1 2 1 3 1 3 <풀이> ( 1 )4

x

2-

y

2=(2

x

)2 -

y

2=(2

x

+

y

)(2

x

-

y

) ( 2 )9

x

2-16

y

2=(3

x

)2-(4

y

)2=(3

x

+4

y

)(3

x

-4

y

) ( 3 )4

x

2-49

y

2=(2

x

)2-(7

y

)2=(2

x

+7

y

)(2

x

-7

y

) ( 4 )9

x

2-64

y

2=(3

x

)2-(8

y

)2=(3

x

+8

y

)(3

x

-8

y

) ( 5 )4

x

2-81

y

2=(2

x

)2-(9

y

)2=(2

x

+9

y

)(2

x

-9

y

) ( 6 )9

x

2-4

y

2=(3

x

)2-(2

y

)2=(3

x

+2

y

)(3

x

-2

y

) ( 7 )16

x

2-25

y

2=(4

x

)2-(5

y

)2=(4

x

+5

y

)(4

x

-5

y

) ( 8 )64

x

2-

y

2=(8

x

)2 -

y

2=(8

x

+

y

)(8

x

-

y

) ( 9 )49

x

2-9

y

2=(7

x

)2-(3

y

)2=(7

x

+3

y

)(7

x

-3

y

) (10)25

x

2-36

y

2=(5

x

)2-(6

y

)2=(5

x

+6

y

)(5

x

-6

y

)

(11)4

x

2- =(2

x

)2-& *2=&2

x

+ *&2

x

- *

(12)9

x

2- =(3

x

)2

-& *2=&3

x

+ *&3

x

- *

(13)25-

x

2=52-&

x

*2 (13)25-

x

2=&5+

x

*&5-

x

* (14)16-

x

2=42-&

x

*2 (14)16-

x

2=&4+

x

*&4-

x

* (15)4

x

2-

y

2=(2

x

)2-&

y

*2 (15)4

x

2-

y

2=&2

x

+

y

*&2

x

-

y

* (16)9

x

2-

y

2=(3

x

)2-&

y

*2 (16)9

x

2-

y

2=&3

x

+

y

*&3

x

-3

y

* 4 3 4 9 16 3 4 9 16 2 3 2 3 4 9 2 3 4 9 1 5 1 5 1 5 1 25 1 4 1 4 1 4 1 16 1 2 1 2 1 2 1 4 1 3 1 3 1 3 1 9

(11)

(17)-16

x

2+ = -16

x

2=& *2-(4

x

)2 (17)-16

x

2+ =& +4

x

*& -4

x

* (18)-

x

2+ = -

x

2=& *2-&

x

*2 (18)-

x

2+ =& +

x

*& -

x

* (19)25

x

2

y

2-49=(5

xy

)2-72=(5

xy

+7)(5

xy

-7) (20)81-36

x

2

y

2=92-(6

xy

)2=(9+6

xy

)(9-6

xy

) 1 3 1 2 1 3 1 2 4 9 4 9 1 3 1 2 1 9 1 4 1 4 1 9 2 3 2 3 4 9 2 3 4 9 4 9 M - 80 ( 1 )(

x

2+5)(

x

2-5) ( 2 )(8+

x

3)(8-

x

3) ( 3 )(

x

+

y

2)(

x

-

y

2) ( 4 )(

x

2+

y

)(

x

2-

y

) ( 5 ) (

x

3+3

y

)(

x

3-3

y

) ( 6 )(

x

+4

y

3)(

x

-4

y

3) ( 7 )(6+

x

2)(6-

x

2) ( 8 )(9+

x

3)(9-

x

3) ( 9 )(2

x

2+7)(2

x

2-7) (10)(8+3

x

3)(8-3

x

3) (11)(4

x

3+

y

4)(4

x

3-

y

4) (12)(5

x

4+2

y

3)(5

x

4-2

y

3) (13)(3

xy

2+4)(3

xy

2-4) (14)(2

x

2

y

+9

z

)(2

x

2

y

-9

z

) (15)(6

xy

2+5

z

2)(6

xy

2-5

z

2) (16)(7

x

2

y

+8

az

2)(7

x

2

y

-8

az

2) <풀이> ( 1 )

x

4-25=(

x

2)2-52=(

x

2+5)(

x

2-5) ( 2 )64-

x

6=82-(

x

3)2=(8+

x

3)(8-

x

3) ( 3 )

x

2-

y

4=

x

2-(

y

2)2=(

x

+

y

2)(

x

-

y

2) ( 4 )

x

4-

y

2=(

x

2)2-

y

2=(

x

2+

y

)(

x

2-

y

) ( 5 )

x

6-9

y

2=(

x

3)2-(3

y

)2=(

x

3+3

y

)(

x

3-3

y

) ( 6 )

x

2-16

y

6=

x

2-(4

y

3)2=(

x

+4

y

3)(

x

-4

y

3) ( 7 )-

x

4+36=36-

x

4=62-(

x

2)2=(6+

x

2)(6-

x

2) ( 8 )-

x

6+81=81-

x

6=92-(

x

3)2=(9+

x

3)(9-

x

3) ( 9 )4

x

4-49=(2

x

2)2-72=(2

x

2+7)(2

x

2-7) (10)64-9

x

6=82-(3

x

3)2=(8+3

x

3)(8-3

x

3) (11)16

x

6-

y

8=(4

x

3)2-(

y

4)2=(4

x

3+

y

4)(4

x

3-

y

4) (12)25

x

8-4

y

6=(5

x

4)2-(2

y

3)2 =(5

x

4+2

y

3)(5

x

4-2

y

3) (13)9

x

2

y

4-16=(3

xy

2)2 -42 =(3

xy

2+4)(3

xy

2-4) (14)4

x

4

y

2-81

z

2=(2

x

2

y

)2-(9

z

)2 =(2

x

2

y

+9

z

)(2

x

2

y

-9

z

) (15)36

x

2

y

4-25

z

4=(6

xy

2)2-(5

z

2)2 =(6

xy

2+5

z

2)(6

xy

2-5

z

2) (16)49

x

4

y

2-64

a

2

z

4=(7

x

2

y

)2-(8

az

2)2 =(7

x

2

y

+8

az

2)(7

x

2

y

-8

az

2) M - 81 ( 1 )3, 16, 3, 4, 3, 4 ( 2 )2(2

x

+5)(2

x

-5) ( 3 )6(

x

+3

y

)(

x

-3

y

) ( 4 )5(3

x

+2

y

)(3

x

-2

y

) ( 5 )

a

(

x

+6)(

x

-6) ( 6 )3

a

(2

x

+3

y

)(2

x

-3

y

) ( 7 )

y

(

x

+

z

)(

x

-

z

) ( 8 )

xy

(

x

+

y

)(

x

-

y

) ( 9 )7(

xy

+2

z

)(

xy

-2

z

)(10)6

ay

(3

x

+1)(3

x

-1) (11)2

x

2

y

(2+3

y

)(2-3

y

)(12)

ax

(2

x

2+

y

)(2

x

2-

y

) (13)5

x

(

xy

+3

z

)(

xy

-3

z

) (14)

xyz

(

x

+

yz

)(

x

-

yz

) <풀이> ( 1 )3

x

2-48=3(

x

2-16) =3(

x

2 -42) =3(

x

+4)(

x

-4) ( 2 )8

x

2-50=2(4

x

2-25) =2{(2

x

)2-52} =2(2

x

+5)(2

x

-5) ( 3 )6

x

2-54

y

2=6(

x

2-9

y

2) =6{

x

2-(3

y

)2} =6(

x

+3

y

)(

x

-3

y

) ( 4 )45

x

2-20

y

2=5(9

x

2 -4

y

2) =5{(3

x

)2-(2

y

)2} =5(3

x

+2

y

)(3

x

-2

y

) ( 5 )

ax

2-36

a

=

a

(

x

2-36) =

a

(

x

2 -62) =

a

(

x

+6)(

x

-6) ( 6 )12

ax

2-27

ay

2=3

a

(4

x

2-9

y

2) =3

a

{(2

x

)2-(3

y

)2} =3

a

(2

x

+3

y

)(2

x

-3

y

) ( 7 )

x

2

y

-

yz

2=

y

(

x

2-

z

2)=

y

(

x

+

z

)(

x

-

z

) ( 8 )

x

3

y

-

xy

3=

xy

(

x

2-

y

2)=

xy

(

x

+

y

)(

x

-

y

) ( 9 )7

x

2

y

2-28

z

2=7(

x

2

y

2-4

z

2) =7{(

xy

)2-(2

z

)2} =7(

xy

+2

z

)(

xy

-2

z

) (10)54

ax

2

y

-6

ay

=6

ay

(9

x

2-1) =6

ay

{(3

x

)2-12} =6

ay

(3

x

+1)(3

x

-1) (11)8

x

2

y

-18

x

2

y

3=2

x

2

y

(4-9

y

2) =2

x

2

y

{22-(3

y

)2} =2

x

2

y

(2+3

y

)(2-3

y

) (12)4

ax

5-

axy

2=

ax

(4

x

4 -

y

2) =

ax

{(2

x

2)2-

y

2} =

ax

(2

x

2+

y

)(2

x

2-

y

) (13)5

x

3

y

2-45

xz

2=5

x

(

x

2

y

2-9

z

2) =5

x

{(

xy

)2-(3

z

)2} =5

x

(

xy

+3

z

)(

xy

-3

z

) (14)

x

3

yz

-

xy

3

z

3=

xyz

(

x

2-

y

2

z

2) =

xyz

{

x

2-(

yz

)2} =

xyz

(

x

+

yz

)(

x

-

yz

)

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