How Do You Calculate the Square Root of a Polynomial?

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To calculate the square root of a polynomial, one must find a polynomial p(x) such that [p(x)]² equals the given polynomial q(x). For the polynomial 4x^4 + 8x^3 + 8x^2 + 4x + 1, it is not possible to find such a p(x) because q must have roots with even multiplicities, which this polynomial does not possess. Attempting to express p as ax² + bx + c and squaring it leads to a system of equations that has no solution. Therefore, finding the square root of this specific polynomial is not feasible. Understanding the conditions for polynomial roots is crucial in these calculations.
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does anyone know any specific method to find the square root of a polynomial like : 4x4 + 8x3 + 8x2 + 4x+ 1 ??
thanks in advance!
 
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You want to find a polynomial p(x) such that [p(x)]2 = q(x)?
If this is it, it is not possible, for a general q, to find p. q must have roots with even multiplicities. This is not the case in your example. Try putting p = ax2+bx+c, square it, and equate coefficients with q. You'll find a system of equations with no solution.
 

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