- #1

chwala

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- Homework Statement
- if## f(x)= x^3+ax^2+bx+c## and the roots of f(x) are ##1, k, k+1##

when ##f(x)## is divided by ##x-2## the remainder is ##20##.

show that ##k^2-3k-18=0##

- Relevant Equations
- factor/remainder theorem

##0=1+a+b+c##

##20=8+4a+2b+c##

it follows that,

##13=3a+b##

and,

##0=k^3+ak^2+bk+c##...1

##0=(k+1)^3+a(k+1)^2+(k+1)b+c##...2

subtracting 1 and 2,

##3k^2+k(3+2a)+14-2a=0##

##20=8+4a+2b+c##

it follows that,

##13=3a+b##

and,

##0=k^3+ak^2+bk+c##...1

##0=(k+1)^3+a(k+1)^2+(k+1)b+c##...2

subtracting 1 and 2,

##3k^2+k(3+2a)+14-2a=0##