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12. Relation between roots and coefficients of any polynomial equation



IN THIS CHAPTER




If α1, α2,α3 ...  αn are the roots of  the equation

f(x)= a0x+a1xn-1  +a2xn-2 +...+an-1x + an =0, then

f(x)= a0 (x-
α1)(x-α2)(x-α3)... (x-αn)

Equating both the RHS terms we get,

a0x+a1xn-1  +a2xn-2 +...+an-1x + an = a0(x-α1)(x-α2)(x-α3)... (x-αn)

Comparing coefficients of
xn-1   on both sides, we get

      S
1 =
α1 + α23 +... + αn = ∑α= -a1/ a0

or, S1= - coeff. of xn-1/coeff. of xn

Comparing coefficients of xn-2   on both sides, we get

      S
2
α1 α2+ α1α3 +...  = ∑αi αj  = (-1)2a2/ a0
                                             i
j
or, S2= (-1)2 coeff. of xn-2/coeff. of xn


Comparing coefficients of xn-3   on both sides, we get

      S
3
α1 α2α3+ α2α3α4 +...  = ∑αi αj αk  = (-1)3a3/ a0
                                                     i
j k
or, S3= (-1)3 coeff. of xn-3/coeff. of xn

  ...       ...      ...     ...    ...     ...    ...    ...

  ...       ...      ...     ...    ...     ...    ...    ...

Sn=α1α2α3... αn =(-1)nan/ a0=   (-1)n constant term/coeff. of xn  


Here,
Sk denotes the sum of the products of the roots taken k at a time.
For example,
S3 denotes the sum of the product of roots taken 3 at a time.

PARTICULAR CASES:

Quadratic Equation:
If
α and β  are roots of the quadratic equation ax2 + bx + c=0, then

α + β = -b/a

α * β = c/a

Cubic Equation:
If
α , β, γ are roots of a cubic equation  ax3 + bx2 + cx + d=0, then

α + β + γ = -b/a

α β +β γ + γα = c/a

α βγ = -d/a


Biquadratic equation :
If α , β, γ, δ are roots of a cubic equation  ax4+ bx3 + cx2 + dx +e=0, then

α + β + γ + δ = -b/a

α β +β γ + γδ +αγ + αδ+ βδ  = c/a

α β γ + αγδ +αβδ + βγδ  = -d/a

αβγδ = e/a






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