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a. \(2a^2+5ab-3b^2-7b-2\)
\(=\left(2a^2+6ab+2a\right)-\left(ab+3b^2+b\right)-\left(2a+6b+2\right)\)
\(=2a\left(a+3b+1\right)-b\left(a+3b+1\right)-2\left(a+3b+1\right)\)
\(=\left(2a-b-2\right)\left(a+3b+1\right)\)
b. \(2x^2-7xy+x+3y^2-3y\)
\(=\left(2x^2-xy\right)-\left(6xy-3y^2\right)+\left(x-3y\right)\)
\(=x\left(2x-y\right)-3y\left(2x-y\right)+\left(x-3y\right)\)
\(=\left(x-3y\right)\left(2x-y\right)+\left(x-3y\right)\)
\(=\left(x-3y\right)\left(2x-y+1\right)\)
c. \(6x^2-xy-2y^2+3x-2y\)
\(=\left(6x^2+3xy\right)-\left(4xy-2y^2\right)+\left(3x-2y\right)\)
\(=3x\left(2x+y\right)-2y\left(2x+y\right)+\left(3x-2y\right)\)
\(=\left(3x-2y\right)\left(2x+y\right)+\left(3x-2y\right)\)
\(=\left(3x-2y\right)\left(2x+y+1\right)\)
a) 4x2 - 5xy + y2 = 4x2 - 4xy - xy + y2 = 4x( x - y ) - y( x - y ) = ( x - y )( 4x - y )
b) x2 - 4xy + 3y2 = x2 - xy - 3xy + 3y2 = x( x - y ) - 3y( x - y ) = ( x - y )( x - 3y )
c) 9x2 + 6xy - 8y2 = 9x2 - 6xy + 12xy - 8y2 = 9x( x - 2/3y ) + 12y( x - 2/3y ) = ( x - 2/3y )( 9x + 12y )
d) 2x2 + 3xy - 5y2 = 2x2 - 2xy + 5xy - 5y2 = 2x( x - y ) + 5y( x - y ) = ( x - y )( 2x + 5y )
e) x2 - 35y2 - 2xy = x2 + 5xy - 7xy - 35y2 = x( x + 5y ) - 7y( x + 5y ) = ( x + 5y )( x - 7y )
f) 2x2 + 10xy + 8y2 = 2( x2 + 5xy + 4y2 ) = 2( x2 + xy + 4xy + 4y2 ) = 2[ x( x + y ) + 4y( x + y ) ] = 2( x + y )( x + 4y )
g) x2 - 10xy + 16y2 = x2 - 2xy - 8xy + 16y2 = x( x - 2y ) - 8y( x - 2y ) = ( x - 2y )( x - 8y )
h) 4x2 + 4xy - 15y2 = 4x2 - 6xy + 10xy - 15y2 = 4x( x - 3/2y ) + 10y( x - 2/3y ) = ( x - 2/3y )( 4x + 10y )
i) -7xy + 3x2 + 2y2 = 3x2 - xy - 6xy + 2y2 = 3x( x - 1/3y ) - 6y( x - 1/3y ) = ( x - 1/3y )( 3x - 6y )
j) 56y2 + 4x2 - 36xy = 4( x2 - 9xy + 14y2 ) = 4( x2 - 2xy - 7xy + 14y2 ) = 4[ x( x - 2y ) - 7y( x - 2y ) ] = 4( x - 2y )( x - 7y )
\(A=x^2-xy+\frac{y^2}{4}+\frac{3}{4}\left(y^2-4y+4\right)+2013\)
\(=\left(x-\frac{y}{2}\right)^2+\frac{3}{4}\left(y-2\right)^2+2013\ge2013\)
\(B\) đề thiếu
\(C\) đề sai, dấu của \(y^2\) là âm thì không tồn tại GTNN
\(P=-\left(x^2-2x+1\right)-\left(4y^2+4y+1\right)+7\)
\(=-\left(x-1\right)^2-\left(2y+1\right)^2+7\le7\)
\(2Q=-4x^2-20y^2+12xy+8x-6y+4\)
\(=-\left(4x^2+9y^2+4-12xy-8x+12y\right)-11\left(y^2-\frac{6}{11}y+\frac{36}{121}\right)+\frac{97}{11}\)
\(=-\left(2x-3y-2\right)^2-11\left(y-\frac{3}{11}\right)^2+\frac{97}{11}\le\frac{97}{11}\)
\(\Rightarrow Q\le\frac{97}{22}\)
Bài 1:
a) 25x2 - 10xy + y2 = (5x - y)2
b) 81x2 - 64y2 = (9x)2 - (8y)2 = (9x - 8y)(9x + 8y)
c) 8x3 + 36x2y + 54xy2 + 27y3
= 8x3 + 27y3 + 36x2y + 54xy2
= (2x + 3y)(4x2 - 6xy + 9y2) + 18xy(2x + 3y)
= (2x + 3y)(4x2 - 6xy + 18xy + 9y2)
= (2x + 3y)(4x2 + 12xy + 9y2)
= (2x + 3y)(2x + 3y)2 = (2x + 3y)3
c) (a2 + b2 - 5)2 - 4(ab + 2)2 = (a2 + b2 - 5)2 - 22(ab + 2)2
= (a2 + b2 - 5)2 - (2ab + 4)2
= (a2 + b2 - 5 - 2ab - 4)(a2 + b2 - 5 + 2ab + 4)
= (a2 - 2ab + b2 - 9)(a2 + 2ab + b2 - 1)
= \(\left [ (a - b)^{2} - 3^{2} \right ]\)\(\left [ (a + b)^{2} - 1\right ]\)
= (a - b - 3)(a - b + 3)(a + b - 1)(a + b + 1)
pn đăng mỗi lần vài bài thôi chứ đăng nhìn ngán lắm
Bài 2:
a) 2x3 + 3x2 + 2x + 3
= 2x3 + 2x + 3x2 + 3
= 2x(x2 + 1) + 3(x2 + 1)
= (x2 + 1)(2x + 3)
b)x3z + x2yz - x2z2 - xyz2
= xz(x2 + xy - xz - yz)
= \(xz\left [ x(x + y) - z(x + y) \right ]\)
= xz(x + y)(x - z)
c) x2y + xy2 - x - y
= xy(x + y) - (x + y)
= (x + y)(xy - 1)
d) 8xy3 - 5xyz - 24y2 + 15z
= 8xy3 - 24y2 - 5xyz + 15z
= 8y2(xy - 3) - 5z(xy - 3)
= (xy - 3)(8y2 - 5z)
e) x3 + y(1 - 3x2) + x(3y2 - 1) - y3
= x3 - y3 + y - 3x2y + 3xy2 - x
= (x - y)(x2 + xy + y2) - 3xy(x - y) - (x - y)
= (x - y)(x2 + xy + y2 - 3xy - 1)
= (x - y)(x2 - 2xy + y2 - 1)
= \((x - y)\left [ (x - y)^{2} - 1 \right ]\)
= (x - y)(x - y - 1)(x - y + 1)
câu f tương tự
a ) \(x^2-x+1\)
\(\Leftrightarrow\left(x^2-2.x.\dfrac{1}{2}+\left(\dfrac{1}{2}\right)^2\right)+\dfrac{3}{4}\)
\(\Leftrightarrow\left(x-\dfrac{1}{2}\right)^2+\dfrac{3}{4}\)
Ta có : \(\left(x-\dfrac{1}{2}\right)^2\ge0\forall x\)
\(\Leftrightarrow\left(x-\dfrac{1}{2}\right)^2+\dfrac{3}{4}\ge\dfrac{3}{4}\)
Vậy GTNN là \(\dfrac{3}{4}\Leftrightarrow x=\dfrac{1}{2}.\)
Giúp e với, tối nay e học r:((