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\(8.\left(3^2+1\right).\left(3^4+1\right)\left(3^8+1\right)\left(3^{16}+1\right)-3^{32}\)
\(=\left(3^2-1\right).\left(3^2+1\right).\left(3^4+1\right)\left(3^8+1\right)\left(3^{16}+1\right)-3^{32}\)
\(=\left(3^4-1\right).\left(3^4+1\right)\left(3^8+1\right)\left(3^{16}+1\right)-3^{32}\)
\(=\left(3^8-1\right)\left(3^8+1\right)\left(3^{16}+1\right)-3^{32}\)
\(=\left(3^{16}-1\right)\left(3^{16}+1\right)-3^{32}=3^{32}-1-3^{32}=-1\)
\(H=\left(2+1\right).\left(2^2+1\right).\left(2^4+1\right).\left(2^8+1\right)-2^{16}=\left(2-1\right).\left(2+1\right).\left(2^2+1\right).\left(2^4+1\right).\left(2^8+1\right)-2^{16}\)
\(=\left(2^2-1\right).\left(2^2+1\right).\left(2^4+1\right).\left(2^8+1\right)-2^{16}=\left(2^4-1\right).\left(2^4+1\right).\left(2^8+1\right)-2^{16}\)
\(=\left(2^8-1\right).\left(2^8+1\right)-2^{16}=\left(2^{16}-1\right)-2^{16}=-1\)
\(\left(2-1\right)\left(2+1\right)\left(2^2+1\right)\left(2^4+1\right)\left(2^8+1\right)\left(2^{16}+1\right)\)
\(=\left(2^2-1\right)\left(2^2+1\right)\left(2^4+1\right)\left(2^8+1\right)\left(2^{16}+1\right)\)
\(=\left(2^4-1\right)\left(2^4+1\right)\left(2^8+1\right)\left(2^{16}+1\right)\)
\(=\left(2^8-1\right)\left(2^8+1\right)\left(2^{16}+1\right)\)
\(=\left(2^{16}-1\right)\left(2^{16}+1\right)\)
\(=2^{32}-1\)
b/ \(100^2+\left(100+3\right)^2+\left(100+5\right)^2+\left(100-6\right)^2\)
\(=100^2+100^2+100^2+100^2+4.100+9+25+36\)
\(=100^2+2.100+1+100^2-4.100+4+100^2-8.100+16+100^2+14.100+49\)
\(=\left(100+1\right)^2+\left(100-2\right)^2+\left(100-4\right)^2+\left(100+7\right)^2\)
a) ( 3x - 1 )2 - 16 = ( 3x - 1 )2 - 42 = ( 3x - 1 - 4 )( 3x - 1 + 4 ) = ( 3x - 5 )( 3x + 3 ) = 3( 3x - 5 )( x + 1 )
b) ( 5x - 4 )2 - 49x2 = ( 5x - 4 )2 - ( 7x )2 = ( 5x - 4 - 7x )( 5x - 4 + 7x ) = ( -2x - 4 )( 12x - 4 ) = -2( x + 2 ).4( 3x - 1 ) = -8( x + 2 )( 3x - 1 )
c) ( 2x + 5 )2 - ( x - 9 )2 = [ ( 2x + 5 ) - ( x - 9 ) ][ ( 2x + 5 ) + ( x - 9 ) ] = ( 2x + 5 - x + 9 )( 2x + 5 + x - 9 ) = ( x + 14 )( 3x - 4 )
d) ( 3x + 1 )2 - 4( x - 2 )2 = ( 3x + 1 )2 - 22( x - 2 )2 = ( 3x + 1 )2 - [ 2( x - 2 ) ]2 = ( 3x + 1 )2 - ( 2x - 4 )2 = [ ( 3x + 1 ) - ( 2x - 4 ) ][ ( 3x + 1 ) + ( 2x - 4 ) ] = ( 3x + 1 - 2x + 4 )( 3x + 1 + 2x - 4 ) = ( x + 5 )( 5x - 3 )
e) 9( 2x + 3 )2 - 4( x + 1 )2 = 32( 2x + 3 )2 - 22( x + 1 )2 = [ 3( 2x + 3 ) ]2 - [ 2( x + 1 ) ]2 = ( 6x + 9 )2 - ( 2x + 2 )2 = [ ( 6x + 9 ) - ( 2x + 2 ) ][ ( 6x + 9 ) + ( 2x + 2 ) ] = ( 6x + 9 - 2x - 2 )( 6x + 9 + 2x + 2 ) = ( 4x + 7 )( 8x + 11 )
f) 4b2c2 - ( b2 + c2 - a2 )2 = ( 2bc )2 - ( b2 + c2 - a2 )2 = [ 2bc - ( b2 + c2 - a2 ) ][ 2bc + ( b2 + c2 - a2 ] = ( 2bc - b2 - c2 + a2 )( 2bc + b2+ c2 - a2 ) = [ a2 - ( b2 - 2bc + c2 ) ][ ( b2 + 2bc + c2 ) - a2 ] = [ a2 - ( b - c )2 ][ ( b + c )2 - a2 ] = ( a - b + c )( a + b - c )( b + c - a )( b + c + a )
g) ( ax + by )2 - ( ay + bx )2
= [ ( ax + by ) - ( ay + bx ) ][ ( ax + by ) + ( ay + bx ) ]
= ( ax + by - ay - bx )( ax + by + ay + bx )
= [ a( x - y ) - b( x - y ) ][ a( x + y ) + b( x + y ) ]
= ( a - b )( x - y )( x + y )( a + b )
h) ( a2 + b2 - 5 )2 - 4( ab + 2 )2
= ( a2 + b2 - 5 )2 - 22( ab + 2 )2
= ( a2 + b2 - 5 )2 - [ 2( ab + 2 ) ]2
= ( a2 + b2 - 5 )2 - ( 2ab + 4 )2
= [ ( a2 + b2 - 5 ) - ( 2ab + 4 ) ][ ( a2 + b2 - 5 ) + ( 2ab + 4 ) ]
= ( a2 + b2 - 5 - 2ab - 4 )( a2 + b2 - 5 + 2ab + 4 )
= [ ( a2 - 2ab + b2 ) - 9 ][ ( a2 + 2ab + b2 ) - 1 ]
= [ ( a - b )2 - 32 ][ ( a + b )2 - 12 ]
= ( a - b - 3 )( a - b + 3 )( a + b - 1 )( a + b + 1 )
i) ( 4x2 - 3x - 18 )2 - ( 4x2 + 3x )2
= [ ( 4x2 - 3x - 18 ) - ( 4x2 + 3x ) ][ ( 4x2 - 3x - 18 ) + ( 4x2 + 3x ) ]
= ( 4x2 - 3x - 18 - 4x2 - 3x )( 4x2 - 3x - 18 + 4x2 + 3x )
= ( -6x - 18 )( 8x2 - 18 )
= -6( x + 3 ).2( 4x2 - 9 )
= -12( x + 3 )( 2x - 3 )( 2x + 3 )
k) 9( x + y - 1 )2 - 4( 2x + 3y + 1 )2
= 32( x + y - 1 )2 - 22( 2x + 3y + 1 )2
= [ 3( x + y - 1 ) ]2 - [ 2( 2x + 3y + 1 ) ]2
= ( 3x + 3y - 3 )2 - ( 4x + 6y + 2 )2
= [ ( 3x + 3y - 3 ) - ( 4x + 6y + 2 ) ][ ( 3x + 3y - 3 ) + ( 4x + 6y + 2 ) ]
= ( 3x + 3y - 3 - 4x - 6y - 2 )( 3x + 3y - 3 + 4x + 6y + 2 )
= ( -x - 3y - 5 )( 7x + 9y - 1 )
l) -4x2 + 12xy - 9y2 + 25
= 25 - ( 4x2 - 12xy + 9y2 )
= 52 - ( 2x - 3y )2
= ( 5 - 2x + 3y )( 5 + 2x - 3y )
m) x2 - 2xy + y2 - 4m2 + 4mn - n2
= ( x2 - 2xy + y2 ) - ( 4m2 - 4mn + n2 )
= ( x - y )2 - ( 2m - n )2
= ( x - y - 2m + n )( x - y + 2m - n )
6) c) x3 - x2 + x = 1
<=> x3 - x2 + x - 1 = 0
<=> (x3 - x2) + (x - 1) = 0
<=> x2 (x - 1) + (x - 1) = 0
<=> (x - 1) (x2 + 1) = 0
=> x - 1 = 0 hoặc x2 + 1 = 0
* x - 1 = 0 => x = 1
* x2 + 1 = 0 => x2 = -1 => x = -1
Vậy x = 1 hoặc x = -1
Bài 5:
a) Đặt \(A=\left(3^2+1\right)\left(3^4+1\right)\left(3^8+1\right)\left(3^{16}+1\right)\)
\(\Rightarrow8A=\left(3^2-1\right)\left(3^2+1\right)\left(3^4+1\right)\left(3^8+1\right)\left(3^{16}+1\right)\)
\(\Rightarrow8A=\left(3^4-1\right)\left(3^4+1\right)\left(3^8+1\right)\left(3^{16}+1\right)\)
\(\Rightarrow8A=\left(3^8-1\right)\left(3^8+1\right)\left(3^{16}+1\right)\)
\(\Rightarrow8A=\left(3^{16}-1\right)\left(3^{16}+1\right)\)
\(\Rightarrow8A=3^{32}-1\)
\(\Rightarrow A=\frac{3^{32}-1}{8}\)
b) (7x+6)2 + (5-6x)2 - (10-12x)(7x+6)
=(7x+6)2 + (5-6x)2 - 2(5-6x)(7x+6)
\(=\left(7x+6-5+6x\right)^2\)
\(=\left(13x+1\right)^2\)
tất cả đống này là hằng đẳng thức : \(a^3+b^3=\left(a+b\right)\left(a^2-ab+b^2\right).\)
\(x^3+1=\left(x+1\right)\left(x^2-x+1\right)\)
\(x^3+4^3=\left(x+4\right)\left(x^2-4x+16\right)\)
\(x^6+2^3=\left(x^2+2\right)\left(x^4-2x^2+4\right)\)
\(\left(3x\right)^3+2^3=\left(3x+2\right)\left(9x^2-6x+4\right)\)
2.Tim x
a,(2x+1)2-4(x+2)2=9
<=> (4x2+4x+1)-4(x2+4x+4)=9
<=> -12x-15=9
<=> -12x=24
<=> x=-2
\(1a,\)\(\left(x^2-0,1\right)=\left(x-\sqrt{0,1}\right)\left(x+\sqrt{0,1}\right)\)
\(1b,\)\(\left(2a^2+b^2\right)^2=\left(2a^2\right)^2+2.2a^2.b^2+\left(b^2\right)^2=4a^4+4a^2b^2+b^4\)
\(1c,\)\(\left(a^2+5\right)\left(5-a^2\right)=\left(5+a^2\right)\left(5-a^2\right)=25-x^4\)
a) \(A=100^2-99^2+98^2-97^2+...+2^2-1^2\)
\(=\left(100-99\right)\left(100+99\right)+\left(98-97\right)\left(98+97\right)+...+\left(2-1\right)\left(2+1\right)\)
\(=100+99+98+97+...+2+1\)
\(=\frac{\left(1+100\right).100}{2}=5050\)
b) \(B=3\left(2^2+1\right)\left(2^4+1\right)...\left(2^{64}+1\right)+1\)
\(=\left(4-1\right)\left(2^2+1\right)\left(2^4+1\right)...\left(2^{64}+1\right)+1\)
\(=\left[\left(2^2-1\right)\left(2^2+1\right)\right]\left(2^4+1\right)...\left(2^{64}+1\right)+1\)
\(=\left(2^4-1\right)\left(2^4+1\right)\left(2^8+1\right).....\left(2^{64}+1\right)+1\)
Cứ tương tự như thế ......
\(B=2^{128}-1+1=2^{128}\)
c) \(C=\left(a+b+c\right)^2+\left(a+b-c\right)^2-2\left(a+b\right)^2\)
\(=a^2+b^2+c^2+2ab+2bc+2ac+a^2+b^2+c^2+2ab-2bc-2ac-2\left(a^2+2ab+b^2\right)\)
\(=2a^2+2b^2+2c^2+4ab-2a^2-4ab-2b^2\)
\(=2c^2\)
Vậy C = 2c2
1. ( 2x + y )( 4x2 - 2xy + y2 ) - 8x3 - y3 - 16
= [ ( 2x )3 + y3 ] - 8x3 - y3 - 16
= 8x3 + y3 - 8x3 - y3 - 16
= -16 ( đpcm )
2. ( 3x + 2y )2 + ( 3x + 2y )2 - 18x2 - 8y2 + 3
= 2( 3x + 2y )2 - 18x2 - 8y2 + 3
= 2( 9x2 + 12xy + 4y2 ) - 18x2 - 8y2 + 3
= 18x2 + 24xy + 8y2 - 18x2 - 8y2 + 3
= 24xy + 3 ( có phụ thuộc vào biến )
3. ( -x - 3 )3 + ( x + 9 )( x2 + 27 ) + 19
= -x3 - 9x2 - 27x - 27 + x3 + 9x2 + 27x + 243 + 19
= -27 + 243 + 19 = 235 ( đpcm )
4. ( x - 2 )3 - x( x + 1 )( x - 1 ) + 13( x - 4 )
= x3 - 6x2 + 12x - 8 - x( x2 - 1 ) + 13x - 52
= x3 - 6x2 + 12x - 8 - x3 + x + 13x - 52
= -6x2 + 26x - 60 ( có phụ thuộc vào biến )
\(\left(2+1\right)\left(2^2+1\right)\left(2^4+1\right)\left(2^8+1\right)\left(2^{16}+1\right)\)
\(=\left(2-1\right)\left(2+1\right)\left(2^2+1\right)\left(2^4+1\right)\left(2^{16}+1\right)\)
\(=\left(2^2-1\right)\left(2^2+1\right)\left(2^4+1\right)\left(2^{16}+1\right)\)
\(=\left(2^4-1\right)\left(2^4+1\right)\left(2^{16}+1\right)\)
\(=\left(2^{16}-1\right)\left(2^{16}+1\right)\)
\(=2^{32}-1\)
\(263^2+74\cdot263+37^2\)
\(=263^2+2\cdot37\cdot263+37^2\)
\(=\left(363+37\right)^2\)
\(=400^2\)