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\(\left(a^2+b^2+ab\right)^2-a^2b^2-b^2c^2-c^2a^2=\left(a^2+b^2+ab-ab\right)\left(a^2+b^2+2ab\right)-c^2\left(a^2+b^2\right)\)
\(=\left(a^2+b^2\right)\left(a+b\right)^2-c^2\left(a^2+b^2\right)=\left(a^2+b^2\right)\left(a+b-c\right)\left(a+b+c\right)\)
\(\left(a+b+c\right)\left(ab+bc+ca\right)-abc\)
\(=\left(a+b+c\right)\left(ab+bc\right)+\left(a+b+c\right)ac-abc\)
\(=\left(ab+b^2+bc\right)\left(a+c\right)+\left(a+c\right)ac+abc-abc\)
\(=\left(a+c\right)\left(ab+b^2+bc+ac\right)\)
\(=\left(a+b\right)\left(b+c\right)\left(c+a\right)\)
câu a (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+c).(a+b-3c)=(a+b+c). (a+b+c+a+b-3c)=(a+b+c).2.(a+b-c)
câu b 4a2b2-(a2+b2-c2) = (2ab-a2-b2+c2).(2ab+a2+b2-c2)
= (c2-(a-b)2).((a+b)2-c2)
= (c-a+b).(c+a-b).(a+b-c).(a+b+c)
câu c a4+b4+c4-2a2b2+2b2c2-2a2c2-4b2c2=(a2-b2-c2)2-4b2c2=(a2-b2-c2-2bc).(a2-b2-c2+2bc)=(a2-(b+c)2).(a2-(b-c)2)=(a-b-c).(a+b+c).(a-b+c).(a+b-c)
câu d dùng pp xét giá trị riêng thay b =c (bạn tự giải ) thì đa thức này nếu coi là đa thức biến b thì đa thức A chia hết cho b-c
a,b,c bình đẳng => A chia hết cho c-a , a-b
=>A= k(a-b)(b-c)(c-a)
thay thử một bộ a,b,c bất kì => k=? (mình đang vội )
thay k tính đc vàoA= k(a-b)(b-c)(c-a)
\(\left(a+b+c\right)\left(ab+bc+ca\right)-abc\)
\(=\left(a+b+c\right)\left(ab+bc\right)+\left(a+b+c\right)ac-abc\)
\(=\left(ab+b^2+bc\right)\left(a+c\right)+\left(a+c\right)ac+abc-abc\)
\(=\left(a+c\right)\left(ab+b^2+bc+ac\right)\)
\(=\left(a+b\right)\left(b+c\right)\left(c+a\right)\)
\(\left(a-b\right)\left(c-a\right)\left(c-b\right)\left(ab+bc+ca\right)\)
\(=a^2b^2\left(a-b\right)+b^2c^2\left(b-a+a-c\right)+c^2a^2\left(c-a\right)\)
\(=a^2b^2\left(a-b\right)+b^2c^2\left(b-a+a-c\right)+c^2a^2\left(c-a\right)\)
\(=a^2b^2\left(a-b\right)+b^2c^2\left(b-a\right)+b^2c^2\left(a-c\right)+c^2a^2\left(c-a\right)\)
\(=b^2\left(a-b\right)\left(a^2-c^2\right)+c^2\left(c-a\right)\left(a^2-b^2\right)\)
\(=b^2\left(a-b\right)\left(a-c\right)\left(a+c\right)+c^2\left(c-a\right)\left(a-b\right)\left(a+b\right)\)
\(=\left(a-b\right)\left(c-a\right)\left[-b^2\left(a+c\right)+c^2\left(a+b\right)\right]\)
\(=\left(a-b\right)\left(c-a\right)\left(-ab^2-b^2c+ac^2+bc^2\right)\)
\(=\left(a-b\right)\left(c-a\right)\left[a\left(c^2-b^2\right)+bc\left(c-b\right)\right]\)
\(=\left(a-b\right)\left(c-a\right)\left[a\left(c-b\right)\left(c+b\right)+bc\left(c-b\right)\right]\)
\(=\left(a-b\right)\left(c-a\right)\left(c-b\right)\left(ab+bc+ca\right)\)
a)x2-2xy+y2+3x-3y-10
=(x2-2xy+y2)+(3x-3y)-10
=(x-y)2+3(x-y)-10
=(x-y).(x-y+3)-10