Sum and product of roots/zeroes
Sum and Product of roots/zeroes of a polynomial-
Standard quadratic equation of a polynomial is ax2+bx+c=0, where a, b,c is constant.
Ans. Let α and β are the roots of the equation x²-2x+3=0
then sum and product of the roots
α+β= - b/a = - coefficient of x/coefficient of x²=-(-2)/1=2
αβ = c/a = constant term/coefficient of x² =3/1=3
then we have to find the equation whose roots are α² and β²
so sum of roots α²+ β² representing in term of α+β and αβ, putting the value of α+β and αβ in this relation
so α²+ β²=(α+β)2 -2αβ=22–2×3=4–6=-2
and product of roots α²× β² already in term of αβ=(αβ)2=32=9
so the equation whose roots are α² and β² is
x²-(α²+ β²)x+α²β²= x²-(-2)x+9
= x²+2x+9 this one is required equation
Standard quadratic equation of a polynomial is ax2+bx+c=0, where a, b,c is constant.
Let α and β
are roots of a polynomial then factors are (x-α) (x-β) = 0
x2-xβ-xα+αβ=0
x2-x(α+β)
+αβ=0
Compare standard
equation of polynomial then
- α+β= - b/a = - coefficient of x/coefficient of x2
Q.1. The roots of the equation
x²-2x+3=0 are α and β. What is the equation whose roots are α², β²?
then sum and product of the roots
α+β= - b/a = - coefficient of x/coefficient of x²=-(-2)/1=2
αβ = c/a = constant term/coefficient of x² =3/1=3
then we have to find the equation whose roots are α² and β²
so sum of roots α²+ β² representing in term of α+β and αβ, putting the value of α+β and αβ in this relation
so α²+ β²=(α+β)2 -2αβ=22–2×3=4–6=-2
and product of roots α²× β² already in term of αβ=(αβ)2=32=9
so the equation whose roots are α² and β² is
x²-(α²+ β²)x+α²β²= x²-(-2)x+9
= x²+2x+9 this one is required equation
