Simplify: ((a^2-b^2)^3+(b^2-c^2)^3+(c^2-...

Simplify: `((a^2-b^2)^3+(b^2-c^2)^3+(c^2-a^2)^3)/((a-b)^3+(b-c)^3+(c-a)^3)`

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`((a^2-b^2)^3+(b^2-c^2)^3+(c^2-a^2)^3)/((a-b)^3+(b-c)^3+(c-a)^3)`
We use the fact that if `a+b+c = 0`, then` a³+b³+ c³ = 3 a b c`.
`(a² - b²) + (b² - c²) + (c² - a²) = 0`
Also,`(a-b) + (b-c)+ (c-a) = 0`
we assume that ` a ≠b ≠ c.`
Hence,
`((a²-b²)³+(b²-c²)³+(c²-a²)³ )/((a-b)³+b-c)³+(c-a)³)`
`=[3 (a² - b²) (b² - c²) (c² - a²) ] / [ 3 (a - b) (b - c) ( c -a) ]`
`=( a+b) (b + c) ( c +a)`

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The value of [(a^2-b^2)^3+(b^2-c^2)^3 + (c^2-a^2)^3] div [(a-b)^3+(b-c)^3+(c-a)^3 ] is equal to: (Given a ne b ne c ) [(a^2-b^2)^3+(b^2-c^2)^3 + (c^2-a^2)^3] div [(a-b)^3+(b-c)^3+(c-a)^3 ] का मान बराबर है: ( a ne b ne c दिया)

If a statement is true for all the values of the variable, such statements are called as identities. Some basic identities are : (1) (a+b)^(2)=a^(2)+2ab+b^(2)=(a-b)^(2)+4ab (3) a^(2)-b^(2)=(a+b)(a-b) (4) (a+b)^(3)=a^(3)+b^(3)+3ab(a+b) (6) a^(3)+b^(3)=(a+b)^(3)=3ab(a+b)=(a+b) (a^(2)-ab) (8) (a+b+c)^(2)=a^(2)+b^(2)+c^(2)+2ab+2bc+2ca=a^(2)+b^(2)+c^(2)+2abc((1)/(a)+(1)/(b)+(1)/(c)) (10) a^(3)+b^(3)+c^(3)-3abc=(a+b+c)(a^(2)+b^(2)+c^(2)-ab-bc-ca) =1/2(a+b+c)[(a-b)^(2)+(b-c)^(2)+(c-a)^(2)] If a+b+c=0,thena^(3)+b^(3)+c^(3)=3abc If a,b, c are real and distinct numbers, then the value of ((a-b)^(3)+(b-c)^(3)+(c-a)^(3))/((a-b).(b-c).(c-a))is

Factorize : (a^(2)-b^(2))^(3)+(b^(2)-c^(2))^(3)+(c^(2)-a^(2))^(3)

Simplify (a+b)(2a-3b+c)-(2a-3b)c

If a^(2)+ b^(2) + c^(2)=0 , then what is ((a^(4)-b^(4))^(3)+(b^(4)-c^(4))^(3)+(c^(4)-a^(4))^(3))/((a^(2)-b^(2))^(3)+(b^(2)-c^(2))^(3)+(c^(2)-a^(2))^(3)) equal to?

What will be factors of (a^(2)-b^(2))^(3) + (b^(2) -c^(2))^(3) + (c^(2) - a^(2))^(3)

The expression (a-b)^(3)+(b-c)^(3)+(c-a)^(3) can be factorized as (a)(a-b)(b-c)(c-a)(b)3(a-b)(b-c)(c-a)(c)-3(a-b)(b-c)(c-a)(d)(a+b+c)(a^(2)+b^(2)+c^(2)-ab-bc-ca)

Prove: |a^3 2a b^3 2b c^3 2c|=2(a-b)(b-c)(c-a)(a+b+c)

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