1.3. 因数分解

次の式を因数分解しなさい。 次の式を因数分解しなさい。

11 6a 6a xx yy -- 15ax15ax yy ++ 9axy9axy

(( )) 22 33 22 22 22 _{(( )) ((22 xx -- yy -- xx +))}22 _{+ yy} 33 x x -- 2727 (( )) 33 _{(( ))44 3x 3x +}22_{+ 5x5x ++ 22} 55 4p 4p -- 12pq12pq ++ 9q9q (( )) 22 22 _{(( ))66 x x -- 2y}22 _{2y -- xy}22 _{xy -- 2x2x -- 5y5y -- 33} 77 2 2 xx -- 11 -- 77 xx -- 11 ++ 66 (( )) (( ))22 _{(( )) (( ))88 6x 6x yy -- 24xy}33 _24xy33

99 3a 3a -- 6ab6ab -- 11a11a ++ 4b4b ++ 66

(( )) 22 _{(( ))1010 a a +}33_{+ 6a6a +}22_{+ 12a12a ++ 88} 11 11 x x ++ 4x4x ++ 1616 (( )) 44 22 _{(( ))1212 10x 10x +}22_{+ 11xy11xy -- 6y6y}22 13 13 a a bb ++ abab ++ bb cc ++ bcbc ++ cc aa ++ caca ++ 2abc2abc (( )) 22 22 22 22 22 22

= = xx -- yy xx -- yy -- 11(( ))(( )) 33 x x -- 2727 == xx -- 33 ((xx ++ 3x3x ++ 99 (( )) 33 _{(( ))} 22 _{)) (( ))44 3x 3x +}22_{+ 5x5x ++ 22 == xx +(( + 11 3x))((3x ++ 22))} 55 4p 4p -- 12pq12pq ++ 9q9q == 2p2p -- 3q3q (( )) 22 22 _{(( ))}22 _{(( ))66 x x -- 2y}22 _{2y -- xy}22 _{xy -- 2x2x -- 5y5y -- 33} = = xx -- yy +22 ₍₍ + 22 xx -- 2y₎₎ 2y -- 5y22 5y -- 33 = = xx -- yy +22 ₍₍ + 22 xx -- ((2y₎₎ 2y +22+ 5y5y ++ 33₎₎ = = xx -- yy +22 ₍₍ + 22 xx -- ((yy +₎₎ + 11 ((2y₎₎ 2y ++ 33₎₎ = = xx +{{ + yy +(( + 11 }} xx -- 2y)) {{ ((2y ++ 33 }})) = = xx +(( + yy ++ 11 xx -- 2y))(( 2y -- 33)) 77 2 2 xx -- 11 -- 77 xx -- 11 ++ 66 (( )) (( ))22 _{(( ))} = = 2A2A -- 7A22 7A ++ 66 = = A((A -- 22 2A))((2A -- 33)) = = xx -- 11 -- 22 2x(( ))((2x -- 22 -- 33)) = = xx -- 33 2x(( ))((2x -- 55)) 88 6x 6x yy -- 24xy24xy (( )) 33 33 = = 6xy6xy((xx -- 4y22 4y22)) = = 6xy6xy xx +(( + 2y2y xx -- 2y))(( 2y))

99 3a 3a -- 6ab6ab -- 11a11a ++ 4b4b ++ 66 (( )) 22 = = 44 -- 6a(( 6a bb +)) + 3a3a -- 11a22 11a ++ 66 = = --22 3a((3a -- 22 bb +)) + aa -- 33 3a(( ))((3a -- 22 )) = = 3a((3a -- 22 aa -- 2b))(( 2b -- 33)) 10 10 a a ++ 6a6a ++ 12a12a ++ 88 == aa ++ 22 (( )) 33 22 _{(( ))}33 11 11 x x ++ 4x4x ++ 1616 (( )) 44 22 = = xx +44+ 8x8x +22+ 1616 -- 4x4x22 = = ((xx +22+ 44 -- 4x₎₎22 4x22 = = ((xx +{{ 22+ 44)) ++ 2x2x}} ((xx +_{{ 22+ 44)) -- 2x2x}} = = ((xx +22+ 2x2x ++ 44 ((xx -- 2x₎₎ 22 2x ++ 44₎₎ 12 12 10x 10x ++ 11xy11xy -- 6y6y == 2x2x ++ 3y3y 5x5x -- 2y2y (( )) 22 22 _{(( ))(( ))} 13 13 a a bb ++ abab ++ bb cc ++ bcbc ++ cc aa ++ caca ++ 2abc2abc (( )) 22 22 22 22 22 22 = = bb +(( + cc aa +)) 22+ ((bb +22+ 2bc2bc ++ cc ))aa +22 + bb cc +22 + bcbc22 = = bb +(( + cc aa +)) 22+ bb +₍₍ + cc aa +₎₎22 + bcbc bb +₍₍ + cc₎₎ = = bb +(( + cc aa +)){{ 22+ bb +₍₍ + cc aa +₎₎ + bcbc}} = = bb +(( + cc aa +))(( + bb aa +))(( + cc)) = = aa +(( + bb bb +))(( + cc cc +))(( + aa)) © オンライン講師ブログ（ © オンライン講師ブログ（https://tutor-blog.comhttps://tutor-blog.com））

次の式を因数分解しなさい。 次の式を因数分解しなさい。 11 x x ++ xx -- 4x4x -- 44 (( )) 33 22 _{(( ))22 6a 6a +}22_{+ 5ab5ab -- 4b4b}22

33 12a 12a -- 16a16a -- 33

(( )) 22 _{(( ))44 8a 8a +}33_{+ bb}33

55 6x 6x -- 17xy17xy ++ 12y12y

(( )) 22 22 _{(( ))66 ((aa +}22_{+ 3a3a -- 22 aa +))}22 22_{+ 3a3a -- 88} 77 aa -- bb yy ++ bxbx -- axax (( )) (( )) (( ))88 1 1 ++ 3ab3ab ++ aa ++ 3b3b 99 a a -- aa bb -- acac ++ bcbc (( )) 33 22 22 22 _{(( ))10 a10 a bb -- cc +}22_{(( ))+ bb cc -- aa +}22_{(( ))+ cc aa -- bb}22_{(( ))} 11 11 2a 2a -- 3ab3ab -- 2b2b -- aa ++ 7b7b -- 33 (( )) 22 22 _{(( ))1212 x x -- 7x}44 _{7x +}22_{+ 1212} 13 13 a a ++ 3a3a ++ 44 (( )) 44 22

=

= xx +(( + 11 ((xx -- 44)) 22 ₎₎

=

= xx +(( + 11 xx +))(( + 22 xx -- 22))(( ))

33 12a 12a -- 16a16a -- 33 == 2a2a -- 33 6a6a ++ 11

(( )) 22 _{(( ))(( )) (( ))44 8a 8a +}33_{+ bb =}33 _{= 2a((2a ++ bb ((4a)) 4a -- 2ab}22 _{2ab ++ bb}22₎₎

55 6x 6x -- 17xy17xy ++ 12y12y == 2x2x -- 3y3y 3x3x -- 4y4y

(( )) 22 22 _{(( ))(( )) (( ))66 ((aa +}22_{+ 3a3a -- 22 aa +))}22 22_{+ 3a3a -- 88} = = AA -- 2A22 2A -- 88 = = A((A ++ 22 A))((A -- 44)) = = ((aa +22+ 3a3a ++ 22 ((aa +₎₎ 22+ 3a3a -- 44₎₎ = = aa +(( + 11 aa +))(( + 22 aa +))(( + 44 aa -- 11))(( )) 77 aa -- bb yy ++ bxbx -- axax (( )) (( )) = = aa -- bb yy +(( )) + bb -- aa xx(( )) = = aa -- bb yy -- aa -- bb xx(( )) (( )) = = aa -- bb yy -- xx(( ))(( )) 88 1 1 ++ 3ab3ab ++ aa ++ 3b3b (( )) = = 3b((3b ++ 11 aa +)) + 3b3b ++ 11 = = 3b((3b ++ 11 aa +))(( + 11)) 99 a a -- aa bb -- acac ++ bcbc (( )) 33 22 22 22 = = ((cc -- aa bb +22 22)) + aa -- ac33 ac22 = = cc +(( + aa cc -- aa bb +))(( )) + aa((aa ++ cc aa -- cc))(( )) = = cc +(( + aa cc -- aa bb -- aa((cc +))(( )) + aa cc -- aa))(( )) = = cc +(( + aa cc -- aa bb -- aa))(( ))(( )) = = bb -- aa aa +(( ))(( + cc cc -- aa))(( )) 10 10 a a bb -- cc ++ bb cc -- aa ++ cc aa -- bb (( )) 22_{(( ))} 22_{(( ))} 22_{(( ))} = = bb -- cc aa -- ((bb -- cc aa +(( )) 22 22 22)) + bb cc -- bc22 bc22 = = bb -- cc aa -- bb +(( )) 22 ₍₍ + cc bb -- cc aa +_{))(( ))} + bcbc bb -- cc_{(( ))} = = bb -- cc aa -- bb +(( )){{ 22 ₍₍ + cc aa +₎₎ + bcbc}} = = bb -- cc aa -- bb aa -- cc(( ))(( ))(( )) = = aa -- bb bb -- cc aa -- cc(( ))(( ))(( )) 11 11 2a 2a -- 3ab3ab -- 2b2b -- aa ++ 7b7b -- 33 (( )) 22 22 = = 2a2a -- 3b22 ₍₍3b ++ 11 aa -- ((2b₎₎ 2b -- 7b22 7b ++ 33₎₎ = = 2a2a -- 3b22 ₍₍3b ++ 11 aa -- bb -- 33 2b_{)) (( ))((}2b -- 11 ₎₎ = = aa -- 2b{{ ((2b -- 11 }} 2a)) {{2a ++ bb -- 33 }}(( )) = = aa -- 2b(( 2b ++ 11 2a))((2a ++ bb -- 33)) 12 12 x x -- 7x7x ++ 1212 (( )) 44 22 = = ((xx -- 44 ((xx -- 3322 ₎₎ 22 ₎₎ = = xx +(( + 22 xx -- 22 ((xx -- 33))(( )) 22 ₎₎ 13 13 a a ++ 3a3a ++ 44 (( )) 44 22 = = aa +44+ 4a4a +22+ 44 -- aa22 = = ((aa +22+ 22 -- aa₎₎22 22 =

= ((aa +22+ 22 ++ aa}} ((aa +22+ 22 -- aa}} =

= ((aa +22+ aa ++ 22 ((aa -- aa +)) 22 + 22))

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次の式を因数分解しなさい。 次の式を因数分解しなさい。 11 a a xx -- yy -- byby ++ bxbx (( )) (( )) (( ))22 a a +33+ 27b27b33 33 6x 6x ++ xx -- 1515 (( )) 22 _{(( ))44 x x +}22_{+ 2y2y -- 3xy}22 _{3xy ++ 5x5x -- 8y8y ++ 66} 55 4x 4x -- 8x8x -- xx ++ 22 (( )) 33 22 _{(( ))66 x x +}44_{+ 2x2x +}22_{+ 99} 77 2 2 ++ 2ab2ab -- 4a4a -- bb

(( )) (( ))88 8x 8x +22+ 2xy2xy -- 15y15y22

99 a a ++ 4a4a -- 55 (( )) 44 22 _{(( ))1010 2abc 2abc -- aa bb -- bb cc -- aa cc -- bc}22 22 22 _{bc +}22_{+ abab +}22_{+ acac}22 11 11 3x 3x ++ 10xy10xy -- 8y8y (( )) 22 22 _{(( ))1212 b b -- ab}33 _{ab -- bc}22 _{bc +}22_{+ acac}22 13 13 ((xx ++ 5x5x ++ 10 10((xx ++ 5x5x ++ 2424 (( )) 22 ₎₎22 22 _{)) (( ))1414 a a -- 3a}33 _{3a +}22_{+ 3a3a -- 11}

= = aa +(( + bb xx -- yy))(( )) 33 6x 6x ++ xx -- 1515 == 2x2x -- 33 3x3x ++ 55 (( )) 22 _{(( ))(( )) (( ))44 x x +}22_{+ 2y2y -- 3xy}22 _{3xy ++ 5x5x -- 8y8y ++ 66} = = xx -- 3y22 ₍₍3y -- 55 xx +₎₎ + 2y2y -- 8y22 8y ++ 66 = = xx -- 3y22 ₍₍3y -- 55 xx +₎₎ + 22((yy -- 4y22 4y ++ 33₎₎ = = xx -- 3y22 ₍₍3y -- 55 xx +₎₎ + 22 yy -- 11 yy -- 33_{(( ))(( ))} = = xx -- 22 yy -- 11 xx -- yy -- 33{{ (( ))}}{{ (( ))}} = = xx -- 2y(( 2y ++ 22 xx -- yy +))(( + 33)) 55 4x 4x -- 8x8x -- xx ++ 22 (( )) 33 22 = = 4x4x xx -- 22 -- xx -- 2222(( )) (( )) = = xx -- 22 ((4x(( )) 4x -- 1122 ₎₎ = = xx -- 22 2x(( ))((2x ++ 11 ((2x)) 2x -- 11)) 66 x x ++ 2x2x ++ 99 (( )) 44 22 = = xx +44+ 6x6x +22+ 99 -- 4x4x22 = = ((xx +22+ 33 -- 4x₎₎22 4x22 = = ((xx +22+ 33 ++ 2x2x ((xx +₎₎ 22+ 33 -- 2x2x₎₎ = = ((xx +22+ 2x2x ++ 33 ((xx -- 2x)) 22 2x ++ 33)) 77 2 2 ++ 2ab2ab -- 4a4a -- bb (( )) = = 2ab2ab -- 4a4a -- bb ++ 22 = = 2a2a bb -- 22 -- bb -- 22(( )) (( )) = = 2a((2a -- 11 bb -- 22))(( ))

88 8x 8x ++ 2xy2xy -- 15y15y (( )) 22 22 = = 2x((2x ++ 3y3y 4x))((4x -- 5y5y)) 99 a a ++ 4a4a -- 55 (( )) 44 22 = = ((aa -- 11 ((aa +22 ₎₎ 22+ 55₎₎ = = aa +(( + 11 aa -- 11 aa +))(( )) 22+ 55 10 10 2abc 2abc -- aa bb -- bb cc -- aa cc -- bcbc ++ abab ++ acac (( )) 22 22 22 22 22 22 =

= --aa bb -- aa cc +22 22 + 2abc2abc ++ abab +22+ acac -- bb cc -- bc22 22 bc22

=

= --((aa bb +22 + aa cc -- 2abc22 2abc -- abab -- ac22 ac +22+ bb cc +22 + bcbc22₎₎

= = -- bb +(( + cc aa -- ((bb +)) 22 22+ 2bc2bc ++ cc ))aa +22 + bcbc bb +₍₍ + cc₎₎ = = -- bb +(( + cc aa -- ((bb +)) 22 + cc)) aa +22 + bcbc bb +₍₍ + cc₎₎ = = -- bb +(( + cc aa -- ((bb +)) 22 + cc))aa ++ bcbc = = -- bb +(( + cc aa -- bb aa -- cc))(( ))(( )) = = aa -- bb bb +(( ))(( + cc cc -- aa))(( )) 11 11 3x 3x ++ 10xy10xy -- 8y8y (( )) 22 22 = = xx +(( + 4y4y 3x))((3x -- 2y2y)) (( ))1212 b b -- abab -- bcbc ++ acac 33 22 22 22 = = acac -- ab22 ab +22+ bb -- bc33 bc22 = = aa((cc -- bb +22 22₎₎+ bb((bb -- cc22 22₎₎ = = aa cc +(( + bb cc -- bb -- bb cc +))(( )) (( + bb cc -- bb))(( )) = = aa -- bb bb +(( ))(( + cc cc -- bb))(( )) © オンライン講師ブログ（ © オンライン講師ブログ（https://tutor-blog.comhttps://tutor-blog.com））

= = A((A ++ 44 A))((A ++ 66)) = = ((xx +22+ 5x5x ++ 44 ((xx +₎₎ 22+ 5x5x ++ 66₎₎ = = xx +(( + 11 xx +))(( + 44 xx +))(( + 22 xx +))(( + 33)) = = xx +(( + 11 xx +))(( + 22 xx +))(( + 33 xx +))(( + 44))

11 6x 6x ++ xyxy -- yy (( )) 22 22 _{(( ))22 2x 2x -- xx -- 8x}33 22 _{8x ++ 44} 33 2x 2x -- 2y2y ++ 3xy3xy ++ 11x11x -- 3y3y ++ 55 (( )) 22 22 _{(( ))44 6a 6a +}22_{+ 23ab23ab -- 48b48b}22 55 2a 2a -- 16b16b (( )) 33 33 _{(( ))66 x x +}44_{+ 4x4x +}22_{+ 1616} 77 2a 2a ++ 3b3b -- 6ab6ab -- 11 (( )) (( ))88 5x 5x +22+ 8x8x -- 44 99 x x -- 6x6x ++ 12x12x -- 88 (( )) 33 22 _{(( ))1010 x x -- xy}22 _{xy -- xx -- yy -- 22} 11 11 ((xx ++ yy -- ((xx -- yy (( )) ))44 ₎₎44 _{(( ))1212 4x 4x +}44_{+ 11} 13 13 xx ++ 11 xx ++ 22 xx ++ 33 xx ++ 44 -- 2424 (( )) (( ))(( ))(( ))(( )) (( ))14 abc14 abc ++ abab ++ bcbc ++ caca ++ aa ++ bb ++ cc ++ 11 © オンライン講師ブログ（ © オンライン講師ブログ（https://tutor-blog.comhttps://tutor-blog.com））

11 6x 6x ++ xyxy -- yy == 2x2x ++ yy 3x3x -- yy (( )) 22 22 _{(( ))(( )) (( ))22 2x 2x -- xx -- 8x}33 22 _{8x ++ 44} = = xx 2x22((2x -- 11 -- 44 2x)) ((2x -- 11)) = = 2x((2x -- 11 ((xx -- 44)) 22 ₎₎ = = 2x((2x -- 11 xx +))(( + 22 xx -- 22))(( )) 33 2x 2x -- 2y2y ++ 3xy3xy ++ 11x11x -- 3y3y ++ 55 (( )) 22 22 = = 2x2x +22+ 3y₍₍3y ++ 99 xx -- 2y₎₎ 2y -- 3y22 3y ++ 55 = = 2x2x +22+ 3y₍₍3y ++ 99 xx -- ((2y₎₎ 2y +22+ 3y3y -- 55₎₎ = = 2x2x +22+ 3y₍₍3y ++ 99 xx -- ((yy -- 11 2y_{)) ))((}2y ++ 55₎₎ = = xx +{{ + 2y((2y ++ 55 2x))}}{{2x -- yy -- 11(( ))}} = = xx +(( + 2y2y ++ 55 2x))((2x -- yy ++ 11)) 44 6a 6a ++ 23ab23ab -- 48b48b == 2a2a -- 3b3b 3a3a ++ 16b16b (( )) 22 22 _{(( ))(( ))} 55 2a 2a -- 16b16b (( )) 33 33 = = 22((aa -- 8b33 8b33)) =

= 22 aa -- 2b(( 2b ((aa +)) 22+ 2ab2ab ++ 4b4b22₎₎

66 x x ++ 4x4x ++ 1616 (( )) 44 22 = = xx +44+ 8x8x +22+ 1616 -- 4x4x22 = = ((xx +22+ 44 -- 4x₎₎22 4x22 = = ((xx +22+ 44 ++ 2x2x ((xx +₎₎ 22+ 44 -- 2x2x₎₎ = = ((xx +22+ 2x2x ++ 44 ((xx -- 2x)) 22 2x ++ 44)) 77 2a 2a ++ 3b3b -- 6ab6ab -- 11 (( )) = = 6ab6ab -- 2a2a ++ 3b3b -- 11 = = 2a2a 3b((3b -- 11 +))+ 3b3b -- 11 = = 2a((2a ++ 11 3b))((3b -- 11)) 88 5x 5x ++ 8x8x -- 44 (( )) 22 = = xx +(( + 22 5x))((5x -- 22)) 99 x x -- 6x6x ++ 12x12x -- 88 == xx -- 22 (( )) 33 22 _{(( ))}33 _{(( ))1010 x x -- xy}22 _{xy -- xx -- yy -- 22} = = --xyxy -- yy ++ xx -- xx -- 2222 = = --yy xx +(( + 11 +)) ((+ xx -- 22 xx +))(( + 11)) = = xx +(( + 11 xx -- yy -- 22))(( )) 11 11 ((xx ++ yy -- ((xx -- yy (( )) ))44 ₎₎44 = = ((xx +{{ + yy +))22+ ((xx -- yy₎₎22_}}{{((xx ++ yy -- ((xx -- yy₎₎22 ₎₎22_}} = = ((2x2x +22+ 2y2y 4xy22₎₎4xy = = 8xy8xy((xx +22+ yy22₎₎ 12 12 4x 4x ++ 11 (( )) 44 = = 4x4x +44+ 4x4x +22+ 11 -- 4x4x22 = = ((2x2x +22+ 11 -- 4x₎₎22 4x22 = = 2x2x +22+ 11 ++ 2x2x 2x2x +22+ 11 -- 2x2x = = ((2x2x +22+ 2x2x ++ 11 ((2x₎₎ 2x -- 2x22 2x ++ 11₎₎

=

= xx((xx ++ 55 ((xx +)) 22+ 5x5x ++ 1010₎₎ == aa +(( + 11 bb +))(( + 11 cc +))(( + 11))

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