Mathematics

Problem

Let

r(x)=x^{3}+kx^{2}+5x+m

be a cubic polynomial. a. (i) Given that

x=1

is a zero of

r(x)

and that the remainder obtained when

r(x)

is divided by

(x+2)

7

, write two independent equations connecting

k

and

m

. b. (ii) Solve these equations to determine the values of

k

and

m

. c. (iii) With the obtained values of

k

and

m

, express

r(x)

in its completely factorised form.