lf α 1 ,α 2,α 3,α 4are the roots of the equation 3x^4−(l+m)x^3+2x+5l=0 and sum of all the roots is equal to 3 and α 1.α2.α3.α4 =10, then (l,m) is equal to
Answers
Answer : (l,m) = (6,3)
Given :-
α 1 ,α 2,α 3,α 4are the roots of the equation 3x^4−(l+m)x^3+2x+5l=0 and
sum of all the roots is equal to 3 and
α 1.α2.α3.α4 =10.
To find :-
Find the value of (l,m) ?
Solution :-
Given bi-quadratic equation is
3x⁴−(l+m)x³+2x+5l=0
On Comparing this with the standard bi quardratic equation ax⁴+bx³+cx²+dx+e = 0
Then we have
a = 3
b= -(l+m)
c = 0
d = 2
e = 5l
And given that
α 1 ,α 2,α 3,α 4 are the roots of the equation
Sum of the roots = -b/a
=> α 1 + α 2 + α 3 + α 4 = -(-(l+m))/3
=> α 1 + α 2 + α 3 + α 4 = (l+m)/3---(1)
According to the given problem
Sum of the roots = 3
=> (l+m)/3 = 3
=> l+m = 3×3
=> l+m = 9 --------(2)
and Product of the roots = e/a
=> α 1.α2.α3.α4 = 5l/3-----(3)
According to the given problem
α 1.α2.α3.α4 =10
=> 5l/3 = 10
=> 5l = 10×3
=> 5l = 30
=> l = 30/5
=> l = 6
Therefore, l = 6
On Substituting the value of l in (2) then
=> 6+m = 9
=> m = 9-6
=> m = 3
Therefore, m = 3
Therefore,
The value of l = 6 and
The value of m = 3
Answer:-
The value of (l,m) for the given problem is (6,3)
Used formulae:-
- The standard bi-quadratic equation is ax⁴+bx³+cx²+dx+e = 0
- Sum of the roots = -b/a
- Product of the roots = e/a