# Algebraic Calculations on Roots

*f*(

*x*) =

*ax*+

^{n}*bx*

^{n}^{-}

^{1}+

*cx*

^{n}^{-}

^{2}+â€¦+K

The sum of the roots taking one at a time = (Coefficient of

*x*

^{n}^{-}

^{1}/Coefficient of

*x*) Ã— (âˆ’1)

^{n}*, where*

^{n}*n*is the number of roots taken at a time.

The sum of the roots taking two at a time = (Coefficient of

*x*

^{n}^{-}

^{2}/Coefficient of

*x*) Ã— (âˆ’1)

^{n}*, where*

^{n}*n*is the number of roots taken at a time.

The sum of the roots taking three at a time = (Coefficient of

*x*

^{n}^{-}

^{3}/Coefficient of

*x*) Ã— (âˆ’1)

^{n}*, where*

^{n}*n*is the number of roots taken at a time.

The product of roots, taking all at a time = (Constant term/Coefficient of

*x*) Ã— (âˆ’1)

^{n}*, where*

^{n}*n*is the total number of roots.

# Quadratic Equation

*f*(

*x*) =

*ax*

^{2}+

*bx*+

*c*= 0

Î± + Î² = (Coefficient of

*x*^{n}^{-}^{1}/Coefficient of*x*) Ã— (âˆ’1)^{n}^{n}^{ }=*x*) Ã— (âˆ’1)

^{n}

^{n}=

# Cubic Equation

*f*(

*x*) =

*ax*

^{3}+

*bx*

^{2}+

*cx*+

*d*= 0

Î± + Î² + Î³ = (Coefficient of

*x*^{n}^{-}^{1}/Coefficient of*x*) Ã— (âˆ’1)^{n}^{n}^{ }=*x*

^{n}^{-}

^{2}/Coefficient of

*x*) Ã— (âˆ’1)

^{n}*=*

^{n}*x*) Ã— (âˆ’1)

^{n}*=*

^{n}

# Bi-quadratic Equation

*f*(

*x*) =

*ax*

^{4}+

*bx*

^{3}+

*cx*

^{2}+

*dx*+

*e*= 0

Î± + Î² + Î³ + Î´ = (Coefficient of

*x*^{n}^{-}^{1}/Coefficient of*x*) Ã— (âˆ’1)^{n}^{n}^{ }=Î±Î² + Î³Î´ + Î±Î´ + Î²Î³ + Î±Î³ + Î´Î² = (Coefficient of

*x*^{n}^{-}^{2}/ Coefficient of*x*) Ã— (âˆ’1)^{n}*=*^{n}Î± Î² Î³ + Î´ Î± Î² + Î³ Î´ Î± + Î² Î³ Î´ = (Coefficient of

*x*^{n}^{-}^{3}/ Coefficient of*x*) Ã— (âˆ’1)^{n}*=*^{n}Î± Î² Î³ Î´ = (Constant term/Coefficient of

*x*) Ã— (âˆ’1)^{n}^{n}=

Example-1

If the polynomial

*ax*^{4}+*bx*^{3}+*cx*^{2}+*dx*+*e*has the property that the product of all the roots, taken at a time, is 1/3rd of the sum of the product of the roots, taking two at*a*time. What is the relationship between*e*and*c*?Solution

The product of all the roots, taken at a time=

The sum of the product of the roots, taking two at a time =

Now, = 1/3

So,

*c*= 3 Ã—*e*Example-2

If

*a*,*b*and*c*are the roots of the equation*x*^{3 }â€“ 3*x*^{2}+ 2*x*+ 1 = 0, then what is the value ofSolution

*ab*+

*bc*+

*ca*= 2 and

*abc*= âˆ’1