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d,=(2bc+b2+c2−a2)(2bc−b2−c2+a2)
=[(b+c)2−a2][−(b+c)2+a2]
=(b+c−a)(b+c+a)2(a−b−c)
\(a^2-b^2+4bc-4c^2\)
\(=a^2-\left(b^2-4bc+4c^2\right)\)
\(=a^2-\left(b-2c\right)^2\)
\(=\left(a-b+2c\right)\left(a+b-2c\right)\)
\(a^2\left(b-c\right)+b^2\left(c-a\right)+c^2\left(a-b\right)\)
\(=a^2\left(b-c\right)+b^2\left(c-a\right)-c^2\left[\left(b-c\right)+\left(c-a\right)\right]\)
\(=a^2\left(b-c\right)+b^2\left(c-a\right)-c^2\left(b-c\right)-c^2\left(c-a\right)\)
\(=\left(b-c\right)\left(a^2-c^2\right)+\left(c-a\right)\left(b^2-c^2\right)\)
\(=\left(b-c\right)\left(a-c\right)\left(a+c\right)+\left(c-a\right)\left(b-c\right)\left(b+c\right)\)
\(=\left(b-c\right)\left(a-c\right)\left(a+c-b-c\right)\)
\(=\left(a-b\right)\left(b-c\right)\left(a-c\right)\)
a) \(=x^2\left(a-b\right)-2\left(a-b\right)=\left(a-b\right)\left(x^2-2\right)\)
b) \(10x^2\left(a-2b\right)^2-\left(x^2+2\right)\left(a-2b\right)^2=\left(a-2b\right)^2\left(10x^2-x^2-2\right)=\left(a-2b\right)^2\left(9x^2-2\right)\)
c) \(50x^2\left(x-y\right)^2-8y^2\left(x-y\right)^2=\left(x-y\right)^2\left(50x^2-8y^2\right)\)
d) \(15a^mb.15^2-15a^mb.3=15^mb\left(15^2-3\right)=15^mb.222\)
a)49-a2+2ab-b2=49-(a2-2ab+b2)=49-(a-b)2=(7-a+b)(7+a-b)
b)a2-b2+4bc-4c2=a2-(b2-2bc+c2)=a2-(b-2c)2=(a-b+2c)(a+b-2c)
c)4b2c2-(b2+c2-a2)2=(4bc-b2-c2+a2)(4bc+b2+c2-a2)
d)(a+b+c)2+(a+b-c)2-4c2
=(a+b+c)2+(a+b-c-2c)(a+b-c+2c)
=(a+b+c)2+(a+b-3c)(a+b+c)
=(a+b+c)(a+b+c+a+b-3c)
=(a+b+c)(2a+2b-2c)
=2(a+b+c)(a+b-c)
a, 49 -a^2 + 2ab - b^2 = 7^2 - ( a^2 - 2ab + b^2) = 7^2 - (a-b)^2 = ( 7-a+b) ( 7 +a -b)
b, a^2 - b^2 + 4bc - 4c^2 = a^2 - ( b^2 - 4bc + 4c^2) = a^2 - ( b - 2c)^2 = ( a - b + 2c)( a + b -2c)
c, 4.b^2.c^2 - ( b^2 + c^2 - a^2)^2 = ( 2bc)^2 - ( b^2 + c^2 - a^2)^2 = ( 2bc - b^2 - c^2 +a^2 )( 2bc + b^2 + c^2 - a^2)
= ( 2bc - b^2 - c^2 +a^2 ) [ (b + c)^2 - a^2) = ( 2bc - b^2 - c^2 +a^2 ) ( b + c -a)(b + c + a)
d, ( a+b+c)^2 +( a + b - c)^2 - 4c^2 = ( a + b + c)^2 + ( a + b - c - 2c)( a + b - c + 2c) = ( a+b +c)^2 + ( a+b - 3c)(a+b+c)
= ( a+b+c)( a+ b+ c + a + b - 3c) = ( a+b+c )( 2a + 2b - 2c) = 2(a+b+c)(a+b-c)