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Prove that a^2+b^2+c^2-a b-b c-c a is al...

Prove that `a^2+b^2+c^2-a b-b c-c a` is always non-negative for all values of `a ,\ b\ a n d\ c`

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To prove that the expression \( a^2 + b^2 + c^2 - ab - bc - ca \) is always non-negative for all values of \( a, b, \) and \( c \), we can follow these steps: ### Step 1: Rewrite the Expression We start with the expression: \[ E = a^2 + b^2 + c^2 - ab - bc - ca \] ...
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RD SHARMA-ALGEBRAIC IDENTITIES-All Questions
  1. Simplify each of the products: (x^3-3x^2-x)(x^2-3x+1)

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  2. Simplify each of the products: (2x^4-4x^2+1)(2x^4-4x^2-1)

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  3. Prove that a^2+b^2+c^2-a b-b c-c a is always non-negative for all valu...

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  4. Write the following in expanded form: (9x+2y+z)^2 (ii) (3x+2y-z)^2

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  5. Write the following in expanded form: (x-2y-3z)^2 (ii) (-x+2y+z)^2

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  6. Write the following in expanded form: (1/4a-1/2b+1)^2 (ii) (-2x+5y-...

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

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

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  9. If a^2+b^2+c^2=20 and a+b+c=0 , find a b+b c+c a

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  10. If a+b+c=9 and a b+b c+c a=40 , find a^2+b^2+c^2 Given:

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  11. If a^2+b^2+c^2=250 and a b+b c+c a=3 , find a+b+c ​

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  12. Write the following in the expanded form: (a+2b+c)^2 (ii) (2a-3b-c)...

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  13. Write the following in the expanded form: (-3x+y+z)^2 (ii) (m+2n-5p...

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  14. Write the following in the expanded form: (2+x-2y)^2 (ii) (a^2+b^2+...

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  15. Write the following in the expanded form: (a b+b c+c a)^2 (ii) (x/y...

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  16. Write the following in the expanded form: (a/(b c)+b/(c a)+c/(a b))^2 ...

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  17. Write the following in the expanded form: (2x-y+z)^2 (ii) (-2x+3y+2...

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

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

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

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