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置きかえの展開・因数分解 同じ部分を探して置き換える練習。 1 置きかえをして展開しなさい。 (1) (x+y+1)(x+y+3) (2) (x+2y-4)(x+2y+4) (3) (x+y+5)(x-y+5) (4) (x-2y+3)(x+y+3) (5) (x+y+1)2 (6) (x-y-1)2 (7) (x+y-2)(x-y+2) (8) (x+2y-1)(x-2y+1) (9) (x2+3xy+y2)(x2-3xy+y2) (10) (x2+2x+3)(x2+2x-4) 2 置き換えをして因数分解しなさい。 (1) x(y+3)+4(y+3) (2) a(b+2)-3(b+2) (3) x(y-z)+5(z-y) (4) x(y-2)-3(2-y) (5) (x+y)2+3(x+y)+2 (6) (x+y) 2+4(x+y)+4 (7) (x-y)2-9 (8) (x+2y)2-4 (9) x(y+2)+2y+4 (10)(x+y)2-6x-6y+9 (11)x2-(y-z)2 (12)x4+3x2-4 置きかえ使った展開 (1) (x+y+1)(x+y+3) =(A+1)(A+3) x+y=A とおく (2) (x+2y-4)(x+2y+4) =(A+4)(A-4) =A2+4A+3 =A2-16 =(x+y)2+4(x+y)+3 =(x+2y)2-16 =x2+2xy+y2+4x+4y+3 =x2+4xy+4y2-16 (3) (x+y+5)(x-y+5) =(A+y)(A-y) x+5=A とおく x+2y=A とおく (4) (x-2y+3)(x+y+3) =(A-2y)(A+y) x+3=A とおく =A2-y2 =A2-Ay-2y2 =(x+5)2-y2 =(x+3)2-(x+3)y-2y2 =x2+10x+25-y2 =x2+6x+9-xy-3y-2y2 (5) (x+y+1)2 =(A+1)2 x+y=A とおく (6) (x-y-1)2 =(A-1)2 x-y=A とおく =A2+2A+1 =A2-2A+1 =(x+y)2+2(x+y)+1 =(x-y)2-2(x-y)+1 =x2+2xy+y2+2x+2y+1 =x2-2xy+y2-2x+2y+1 (7) (x+y-2)(x-y+2) =(x+A)(x-A) y-2=A とおく (8) (x+2y-1)(x-2y+1) =(x+A)(x-A) =x2-A2 =x2-A2 =x2-(y-2)2 =x2-(2y-1)2 =x2-y2+4y-4 =x2-4y2+4y-1 (9) (x2+3xy+y2)(x2-3xy+y2) =(A+3y)(A-3y) =A2-9x2y2 x2+y2=A とおく =(x2+y2)2-9x2y2 (10) 2y-1=A とおく (x2+2x+3)(x2+2x-4) =(A+3)(A-4) x2+3=A とおく =A2-A-12 =(x2+2x)2-(x2+2x)-12 =x4+2x2y2+y4-9x2y2 =x4+4x3+4x2-x2-2x-12 =x4-7x2y2+y4 =x4+4x3+3x2-2x-12 置きかえ使った因数分解 (1) x(y+3)+4(y+3) =xA+4A (2) a(b+2)-3(b+2) y+3=A とおく =aA-3A =A(x+4) =A(a-3) =(y+3)(x+4) =(b+2)(a-3) (3) x(y-z)+5(z-y) (4) x(y-2)-3(2-y) =x(y-z)-5(-z+y) =xA-5A y-z=A とおく =x(y-2)+3(-2+y) =xA+3A =A(x-5) =A(x+3) =(y-z)(x-5) =(y-2)(x+3) (5) (x+y)2+3(x+y)+2 =A2+3A+2 x+y=A とおく =A2+4A+4 =(A+2)2 =(x+y+1)(x+y+2) =(x+y+2)2 =A2-9 x-y=A とおく y-2=A とおく (6) (x+y) 2+4(x+y)+4 =(A+1)(A+2) (7) (x-y)2-9 b+2=A とおく x+y=A とおく (8) (x+2y)2-4 =A2-4 x+2y=A とおく =(A+3)(A-3) =(A+2)(A-2) =(x-y+3)(x-y-3) =(x+2y+2)(x+2y-2) (9) x(y+2)+2y+4 (10)(x+y)2-6x-6y+9 =(x+y)2-6(x+y)+9 =x(y+2)+2(y+2) =xA+2A y+2=A とおく =A2-6A+9 =A(x+2) =(A-3)2 =(y+2)(x+2) =(x+y-3)2 (11)x2-(y-z)2 =x2-A2 y-z=A とおく x+y=A とおく (12)x4+3x2-4 =A2+3A-4 =(x+A)(x-A) =(A+4)(A-1) =(x+y-z)(x-y+z) =(x2+4)(x2-1) =(x2+4)(x+1)(x-1) x2=A とおく