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Using Duplexes in Squaring Polynomials

What is the coefficient of \(x^2\) in the expansion of \((3x^2 + 2x + 4)^2\)?

This question, given in the 4th MATH-Inic Vedic Mathematics National Challenge can be easily solved using duplexes.  Following the procedure discussed in an article in the previous issue of our MATH-Inic Newsletter, https://www.math-inic.com/blog/using-duplexes/,

\((3x^2+2x+4)^2\)

= D\((3x^2)\) + D\((3x^2+2x)\) + D\((3x^2+2x+4)\) + D\((2x+4)\) + D\((4)\)

 = \((3x^2)^2 + 2(3x^2)(2x) + [2(3x^2)(4)+(2x)^2] + 2(2x)(4) + 4^2\)

 = \(9x^4 + 12x^3 + 28x^2 + 16x + 16\)

However, the question only asks for the coefficient of \(x^2\). We note that the 3rd duplex will give the \(x^2\) term thus solving only for the 3rd term will suffice:

D\((3x^2 + 2x + 4) = 2(3x^2)(4) +  2x^2 = 24x^2 + 4x^2 = 28x^2\)

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