Given that zeroes of cubic polynomial `x^(3)-6x^(2)+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 as zeroes of the given polynomial.

Answer

N/a

Explanation

Let `" "p(x)=x^(3)-6x^(2)+3x+10`
Here, A=1, B=-6, C=3, D=10
Let`" " alpha=a, beta=a+b, gamma=a+2b`
Now, `" " alpha+beta+gamma=-(B)/(A)`
`implies a+a+b+a+2b=-(-6)/(1)`
`implies " "3a+2b=6`
`impleis " " a+b=2 " "`..............(1)
and `" " alphabetagamma=(-D)/(A) `
`implies " " a(a+b)(a+2b)=-(10)/(1)`
`implies " "a(2){a+2(2-a)}=-(10)/(1)`
`implies " " a(2){a+2(2-a)}=-10 " "`[from (1)]
`implies " "a(4-a)=-5`
`implies" "4a-a^(2)=-5`
`implies " "a^(2)-4a-5=0`

`implies" "a^(2)-5a+a-5=0`
`implies " "a(a-5)+1(a-5)=0`
`implies " "(a-5)(a+1)=0`
`implies " " a-5=0 " " or " " a+1=0` ,
`a=5 " "or " "a=-1`
a=5, then from equation (1)`" " b=2-5=-3`
a=-1, then from equation (1)`" " b=2-(-1)=3`
When a=5, b=-3, then
`alpha=a=5, beta=a+b=5-3=2, gamma=a+2b=5-6=-1`
When a=-1,b=3, then
`alpha=a=-1,betaa+b=-1+3=2, gamaa=alpha+2b=-1+6=5`
`:.`Roots are 5,2,-1.`" "` Ans.