Given that the zeroes of the cubic polynomial x3 – 6x2 + 3x + 10 are of the form a,a + b, a + 2b for some real numbers a and b, find the values of a and b as well asthe zeroes of the given polynomial.
Given that the zeroes of the cubic polynomial x3 – 6x2 + 3x + 10 are of the form a,a + b, a + 2b for some real numbers a and b, find the values of a and b as well asthe zeroes of the given polynomial.
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Answer:
a=-1,b= 3
Step-by-step explanation:
given equn is
x³-6x²+3x+10
given data that the roots are in AP
so by using elegant rule
let us assume the roots are
p-q ,p ,p+q
now
use the relationship between roots and coefficients
so
S1 = α+β +γ = -B/A
=> p-q+p+p+q=6
3p=6
p=2 ............ important one
then after
S3= αβγ= -D/A
=> p(p+q)(p-q)= -10
=> p(p²-q²)=-10
=> 2(4-q²) = -10
(4-q²) = -5
q² = 9
q=±3
now
you can choose any one value for 'q' as it gives same ans
so
p-q = -1 = a ..........[ as per given condition]
p= 2 = a+b
p+q = 5 = a+2b
so
put value of 'a' in any one given condition
so
and solve it
a+b= 2
b= 2 +1
b = 3
therefore
the roots of cubic polynomial is
-1,2,5
and the values of a and b are
-1 ,3