This is a direct method to find the roots of any polynomial equation with real coefficients. The basic idea behind this method is to separate the roots of the equations by squaring the roots. This can be done by separating even and odd powers of x in 

and squaring on both sides. Thus we get,

substituting y for -x² we have 

where 

b₁ = a₁² - 2a₂ 

Thus all the b_i's ( i = 0, 1, 2, . . . n) are known in terms of a_i's. The roots of this equation are -s₁²,. -s₂², . . ., s_n² where s₁, s₂ ,  .  .  ., s_n are the roots of P_n (x) = 0.

A typical coefficient b_k of  b_i , i = 1, 2, . . .n  is obtained by following 

The terms alternate in sign starting with a +ve sign. The first term is the square of the coefficient a_k . The second term is twice the product of the nearest nebhouring coefficients a_i-1 and a_i+1. The third is twice the product of the next nebhouring coefficients a_i-2 and a_i+2. This procedure is continued until there are no available coefficients to form the cross products.

This procedure can be repeated many times so that the final equation

has the roots R₁, R₂ , .  .  ., R_n such that

if we repeat the process for m times.

If we assume |s₁| > |s₂| > . . . |s_n|     then    |R₁| >> |R₂| >> . . .>> |R _n|

that is the roots R_i are very widely separated for large m.

Now we have 

which gives        R_i = -B_i / B _i-1 ,              i = 1, 2, . . . n

where B₀ = 1 .

since                 | s_i |^{2^m} = | R_i |          i = 1, 2, . . . n

Þ     | s_i | = | R_i |^{2^-m}          i = 1, 2, . . . n

This determines the absolute values of the roots and substitution in the original equation will give the sign of the roots.

Example : 

Find the roots of   x³ -7x² +14x - 8 = 0

a[]                1          -7               14                  -8 

b[]                1          21               84                  64 

roots =         4.583             2               0.873

b[]                1         273              4368            4096 

roots =         4.065             2               0.984

b[]                1         65793          1.68E7         1.68E7 

roots =         4.002             2               0.9995

Thus the absolute values of the roots are 4, 2, 1. 

Since f(1) = 0, f(2) = 0 and f(4) = 0, the signs of the roots 1, 2 and  4 are all positive.

Work out here :