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Prove that (a+b+c)^(3)-a^(3)-b^(3)-c^(3)...

Prove that `(a+b+c)^(3)-a^(3)-b^(3)-c^(3)=3(a+b)(b+c)(c+a)`.

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To prove that \((a+b+c)^3 - a^3 - b^3 - c^3 = 3(a+b)(b+c)(c+a)\), we will start by expanding the left-hand side and simplifying it step by step. ### Step 1: Expand the left-hand side We begin with the expression \((a+b+c)^3\). Using the binomial expansion formula, we can expand this as follows: \[ (a+b+c)^3 = a^3 + b^3 + c^3 + 3(a+b)(b+c)(c+a) \] ...
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NAGEEN PRAKASHAN-POLYNOMIALS-Problems From NCERT/exemplar
  1. Find the value of the polynomial 5x-4x^(2)+3 at : (i)x=0 " " ...

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  2. Find the remainder when x^(3)+3x^(2)+3x+1 is divided by : (i) x+1 " ...

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  3. Find the remainder when x^(3)-ax^(2)+6x-a is divided by x-a.

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  4. Find the value of k, if x-1 is a factor of p(x) in each of the followi...

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  5. Factorise the following using appropriate identities : (i) 9x^(2)+6x...

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  6. Give possible expressions for the length and breath of each of the fol...

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  7. What are possible expressions for the dimensions of the cuboids whose ...

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  8. Factorise the following : (i) 4x^(2)+20x+25 " " (ii) ...

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  9. Without finding the cubes, factorise : (x-2y)^(3)+(2y-3z)^(3)+(3z-x)...

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  10. Find the value of : (i) x^(3)+y^(3)-12xy+64, when x+y=-4 (ii) x^(3...

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  11. Facorise the following (i) 9x^(2)+4y^(2)+16z^(2)+12xy-16yz-24xz (...

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  12. Without actual division, prove that 2x^(4)-5x^(3)+2x^(2)-x+2 is divisi...

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  13. Simplify (2x-5y)^(3)-(2x+5y)^(3).

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  14. Multiply x^(2)+4y^(2)+z^(2)+2xy+xz-2yz by (-z+x-2y)

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  15. If a, b, c are all non-zero and a+b+c=0, then (a^(2))/(bc)+(b^(2))/(ca...

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  16. Prove that (a+b+c)^(3)-a^(3)-b^(3)-c^(3)=3(a+b)(b+c)(c+a).

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  17. If a+b+c=5 and ab+bc+ca=10, then prove that a^(3)+b^(3)+c^(3)-3abc=-25...

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  18. The polynomial p(x)=x^(4)-2x^(3)-ax+3a-7 when divided by x+1 leaves re...

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