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Algebraic Calculations on Roots

f(x) = axn + bxn-1 + cxn-2 +…+K

The sum of the roots taking one at a time = (Coefficient of xn-1/Coefficient of xn) × (−1)n, where n is the number of roots taken at a time.

The sum of the roots taking two at a time = (Coefficient of xn-2/Coefficient of xn) × (−1)n, where n is the number of roots taken at a time.

The sum of the roots taking three at a time = (Coefficient of xn-3/Coefficient of xn) × (−1)n, where n is the number of roots taken at a time.

The product of roots, taking all at a time = (Constant term/Coefficient of xn) × (−1)n, where n is the total number of roots.

Quadratic Equation

f(x) = ax2 + bx + c = 0
 
Assume that the roots are α and β.
α + β = (Coefficient of xn-1/Coefficient of xn) × (−1)n Description: 2176.png
 
α β = (Constant term/Coefficient of xn) × (−1)n 
Description: 2185.png

Cubic Equation

f(x) = ax3 + bx2 + cx + d = 0
 
Assume that the roots are α, β and γ.
α + β + γ = (Coefficient of xn-1/Coefficient of xn) × (−1)n Description: 2194.png
 
α β + γ α + β γ = (Coefficient of xn-2/Coefficient of xn) × (−1)n = Description: 2203.png
 
α β γ = (Constant term/Coefficient of xn) × (−1)n  Description: 2212.png

Bi-quadratic Equation

f(x) = ax4 + bx3 + cx2 + dx + e = 0
 
Assume that the roots are α, β, γ and δ.
α + β + γ + δ = (Coefficient of xn-1/Coefficient of xn) × (−1)n Description: 2221.png
 
αβ + γδ + αδ + βγ + αγ + δβ = (Coefficient of xn-2/ Coefficient of xn) × (−1)n = Description: 2230.png
 
α β γ + δ α β + γ δ α + β γ δ = (Coefficient of xn-3/ Coefficient of xn) × (−1)n = Description: 2239.png
 
α β γ δ = (Constant term/Coefficient of xn) × (−1)n 
Description: 2243.png
 
Example-1
If the polynomial ax4 + bx3 + cx2 + 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=Description: 2247.png
The sum of the product of the roots, taking two at a time =Description: 2256.png
Now, Description: 2265.png= 1/3Description: 2269.png
So, c = 3 × e
 
Example-2
If ab and c are the roots of the equation x3 – 3x2 + 2x + 1 = 0, then what is the value of Description: 2273.png
Solution
Description: 2283.png
 
ab + bc + ca = 2 and abc = −1
 
Description: 2292.png





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