5th Row Coefficients, 6th Row Terms

Algebra Level 5

\[x^6+5x^5y+10x^4y^2+kx^3y^3+10x^2y^4+5xy^5+y^6\ge 0,\]

Find the absolute value of the smallest possible value for \(k\), such that the inequality above is true for all non-negative reals \(x\) and \( y \).

Details and Assumptions

  • You may use the following algebraic identities. \((x+y)^5=x^5+5x^4y+10x^3y^2+10x^2y^3+5xy^4+y^5\)
    \((x+y)^6=x^6+6 x^5 y+15 x^4 y^2+20 x^3 y^3+15 x^2 y^4+6 x y^5+y^6\)

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