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1. x2 + 2xy + y2 - xz - yz
= ( x2 +2xy + y2 ) - z ( x + y )
= ( x + y )2 - z ( x + y )
= ( x + y ) [( x + y ) - z ]
= ( x + y ) ( x + y - z )
1 x^2+2xy+y^2-xz-yz
=(x+y)^2-z(x+y)
=(x+y)(x+y-z)
2 (7x^2-14xy+7^2)-29z^2
=7(x^2-2xy+1)-29z^2
=7(x-1)^2-29z^2
=7(x-1)^2-25z^2-7z^2
=7(x-1-5)(x-1+5)-7z^2
=7(x-6)(x+4)-7z^2
=7((x+6)(x+4)-z^2)
3 5x^3-5x^2y+10x^2-10xy
=5x(x^2-xy+2x-2y)
4 5x^2-10xy+5y^2-20z^2
=5(x^2-2xy+y^2)-20z^2
=5(x+y)^2-20z^2
=5((x+y)^2-4z^2)
=5((x+y-2z)(x+y+2z))
a) \(x^2-y^2-2x-2y=\left(x-y\right)\left(x+y\right)-2\left(x+y\right)=\left(x+y\right)\left(x-y-2\right)\)
b) \(18m^2-36mn+18n^2-72p^2=18\left(m^2-2mn+n^2-4p^2\right)=18\left[\left(m-n\right)^2-4p^2\right]\\ =18\left(m-n+2p\right)\left(m-n-2p\right)\)
c) \(2x^2-5x+7=2x^2+2x-7x-7=2x\left(x+1\right)-7\left(x+1\right)=\left(x+1\right)\left(2x-7\right)\)
d) \(\left(x-1\right)\left(x-2\right)\left(x-3\right)\left(x-4\right)-24\)
\(=\left[\left(x-1\right)\left(x-4\right)\right]\left[\left(x-2\right)\left(\cdot x-3\right)\right]-24\)
\(=\left(x^2-5x+4\right)\left(x^2-5x+6\right)-24\)
Đặt \(x^2+5x+5=t\) pt trở thành:
\(\left(t-1\right)\left(t+1\right)-24=t^2-1-24=t^2-25=\left(t-5\right)\left(t+5\right)\)
Thay vào bên trên
1) x(x+3)(x+1)(x+2)+1= (x^2 +3x)(x^2+3x+2)+1
Đặt x^2+3x=a ta có:
a(a+2)+1= a^2 +2a +1= (a+1)^2
Trở về ẩn x có
x(x+3)(x+1)(x+2)+1= (x^2 +3x)^2=x^2(x+3)^2
2) Đặt x^2 + x=a, ta có
a^2 +3a +2= (a^2+a) + (a+2)=a(a+2) +(a+2)=(a+1)(a+2)
Trở về ẩn x có
BT=( x^2 + x+1)(x^2 + x+2)
3) BT= (x-y)^2 +3(x-y) -10
đặt x-y=a ta có
a^2+3a -10= (a^2-2a)+(5a-10)=a(a-2)+5(a-2)=(a+5)(a-2)
trở về ẩn x,y có
BT= (x-y +5)(x-y-2)
- 9y3x – 36y2x= 9xy(y2–4) =9xy(y–4)(y+4)
- 64-y^2 – x^2 – 2xy = 64– (x^2 + 2xy + y^2) = 82 – (x+y)2 = (8 – x –y)(8+x+y)
- x^2 + x – 30= x^2 + x – 25 – 5 = (x2 – 25)(x – 5)= (x-5)(x+5)(x-5)= (x-5)^2 (x+5)
+,
= (x-y)^2 - z.(x-y) = (x-y).(x-y-z)
+,
=(x-y).(x+y)-(x-y) = (x-y).(x+y-1)
+,
=x^3.(x-1)-(x^2-1) = x^3.(x-1).(x-1).(x+1) = (x-1).(x^3-x-1)
+,
=a.(x^2-y^2)-7.(x+y) = a.(x+y).(x-y)-7.(x+y) = (ax+ay-7).(x+y)
\(x^2-2xy+y^2-xz+yz\)
\(=\left(x-y\right)^2-\left(xz-yz\right)\)
\(=\left(x-y\right)\left(x-y\right)-z\left(x-y\right)\)
\(=\left(x-y\right)\left(x-y-z\right)\)
\(x^2-y^2-x+y\)
\(=\left(x-y\right)\left(x+y\right)-\left(x-y\right)\)
\(=\left(x-y\right)\left(x+y-1\right)\)
\(x^4-x^3-x^2+1\)
\(=x^3\left(x-1\right)-\left(x^2-1\right)\)
\(=x^3\left(x-1\right)-\left(x-1\right)\left(x+1\right)\)
\(=\left(x-1\right)\left(x^3-x-1\right)\)
\(ax^2-ay^2-7x-7y\)
\(=a\left(x^2-y^2\right)-\left(7x+7y\right)\)
\(=a\left(x-y\right)\left(x+y\right)-7\left(x-y\right)\)
\(=\left(x-y\right)\left[a\left(x+y\right)-7\right]\)
\(1,\)\(3x-3y-x^2+2xy-y^2\)
\(=-\left(x^2-2xy+y^2\right)+\left(3x-3y\right)\)
\(=-\left(x-y\right)^2+3\left(x-y\right)\)
\(=\left(x-y\right)\left[-\left(x-y\right)+3\right]\)
\(=\left(x-y\right)\left(3-x+y\right)\)
\(2,\)\(49\left(x-4\right)^2-9\left(x+2\right)^2\)
\(=\left[7\left(x-4\right)\right]^2-\left[3\left(x-2\right)\right]^2\)
\(=\left(7x-28-3x+6\right)\left(7x-28+3x-6\right)\)
\(=\left(4x-22\right)\left(10x+34\right)\)
\(3,\)\(x^4+4x^2-5\)
\(=x^4-x^2+5x^2-5\)
\(=x^2\left(x^2-1\right)+5\left(x^2-1\right)\)
\(=\left(x^2+5\right)\left(x-1\right)\left(x+1\right)\)
a)
\(10x^2+10xy+5x+5y\)
\(=10x\left(x+y\right)+5\left(x+y\right)\)
\(=5\left(x+y\right)\left(2x+1\right)\)
b)
\(x^3+x^2-x-1\)
\(=x^2\left(x+1\right)-\left(x+1\right)\)
\(=\left(x-1\right)\left(x^2-1\right)\)
\(=\left(x-1\right)^2\left(x+1\right)\)
c)
\(x+2a\left(x-y\right)-y\)
\(=\left(x-y\right)+2a\left(x-y\right)\)
\(=\left(x-y\right)\left(2a+1\right)\)
d)
\(x^2-y^2+7x-7y\)
\(=\left(x+y\right)\left(x-y\right)+\left(x-y\right)\)
\(=\left(x-y\right)\left(x+y+1\right)\)