Given fhat the zeroes of the cubic polynomial x^3 - 6x^3 + 3x+10 are of tge form a,a+b,a+2b for some real number a and b . Find the values of a and b asa well as the zeroes of following polynomial
Answers
Given
a, a+b, a+2b are roots or zeroes of cubic polynomial
x³-6x²+3x+10=0
From this cubic polynomial,
Sum of the zeroes or roots are given as,
--> a+2b+a+a+b = -a of x²/ a of x³
where a is coefficient of x² and x³
-->3a+3b = -(-6)/1 = 6
--> 3(a+b) = 6
-->a+b = 2 --------- 1)
--> b = 2-a
Product of roots/zeroes--> (a+2b)(a+b)a = -constant/coefficient of x³
-->(a+b+b)(a+b)a = -10/1
Put the value of a+b=2 ,
(2+b)*(2)a = -10
(2+b)*2a = -10
(2+2-a)*2*a = -10
(4-a)*2*a = -10
4*a-a² = -5
a²-4*a-5 = 0
a²-5*a+a-5 = 0
(a-5)(a+1) = 0
a-5 = 0 or a+1 = 0
either or
a = 5 a = -1
a = 5, -1 ---> 1)
a+b = 2
or when, a = 5,
5+b=2
-->b=-3
a = -1, -1+b=2 ⇒ b= 3
so that If a=5 then the value of b= -3
or
so that If a= -1 then the value of b=3