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Question

Given that the zeros of the cubic polynomial x36x2+3x+10 are of the form a,a+b and a+2b for some real numbers a and b, find the values of a and b

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Solution

Given that a,a+b &a+2b are roots of given polynomila x36x2+3x+10
Sum of the roots =a+2b+a+a+b=coefficient of x2coefficient of x3
3a+3b=(6)1=6
a+b=2.......(1) b=2a
Product of roots =(a+2b)(a+b)a=constantcoefficient of x3
(a+b+b)(a+b)a=101=10
(2+b)(2)a=10
(2+2a)2a=10
(4a)2a=10
4aa2=5
a24a5=0
a25a+a5=0
(a5)(a+1)=0
a=5 or a=1
when a=5, 5+b=2b=3
a=1, 1+b=2b=3

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