square root of  a fourth degree polynomial which is a perfect square
x^4 -6(x^3) + 19(x^2) -30x + 20 using  long division method




find the square root of x^4 -6(x^3) + 19(x^2) -30x + 20

square root of a fourth
              degree perfect square polynomial
first split into groups of 2 terms each from the last term

x^4 + [ -6(x^3) + 19(x^2)] + [ -30x + 20]

now first term is x^4 which is a perfect square x^4 = [x^2]^2

using long division method

first step gives remainder 0 then the next group [ -6(x^3) + 19(x^2)]
is taken down

current quotient is [x^2] multiply it by 2 get 2[x^2]

so the the next divisor is  of the form 2[x^2] + ?

for getting the unknown term ? divide the leading term in the group
-6(x^3) with 2[x^2]  to get -3x

so the new divisor is 2[x^2] -3x

introduce  (-3x) in the quotient.

multiply{2[x^2] -3x}*{-3x} to get -6[x^3] +9x

subtract to get 10[x^2]
bring down the next group to get

10[x^2] -30x +20

multiply the current quotient [x^2] - 3x with 2

to get 2[x^2] - 6x

so new divisor is of the form 2[x^2] - 6x + ?

divide the leading terms  10[x^2] with 2[x^2] to get 5

so new divisor is 2[x^2] - 6x +5

introduce +5 in the quotient
 multiply 2[x^2] - 6x +5 with 5 to get 10[x^2] - 30x + 25

subtract to get the remainder 0

and the square root | [x^2] -3x +5 |

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