Mathematics

Problem

The cubic polynomial

Q(x)=x^{3}+ax^{2}+bx+c

satisfies all the following conditions: (i) It is divisible by

x-2

. (ii) It leaves a remainder of

10

when divided by

x+1

. (iii) The sum of its coefficients equals

3

. (a) Translate each condition (i)–(iii) into an algebraic equation involving

a

b

and

c

. (b) Solve the system of equations obtained in part (a) to find the values of

a

b

and

c

. (c) Using the values found, verify explicitly that all three given conditions hold for

Q(x)