POLYNOMIALS A1

When mx³+nx²-18x-45 is divided by x²-9 the remainder is zero. Calculate the values of m and n.

solution

x²-9 = 0

(x)²-(3)²=0

(x+3)(x-3) = 0

x+3= 0  OR  x-3 = 0

x= 3  OR  x= -3

When x=3,

 m(3)³+n(3)²-18(3)-45=0

27m + 9n -54 -45 = 0

27m + 9n -99 = 0

27m + 9n = 99  …………………..(1)

When x=-3,

 m(-3)³+n(-3)²-18(-3)-45=0

-27m + 9n + 54 - 45 = 0

-27m + 9n + 9 = 0

-27m + 9n = -9 ………………………………(2)

Solving 1 and 2 simultaneously,

  27m + 9n = 99  …………………..(1)

-27m + 9n = -9 ………………..……(2)

By using elimination method,

      27m + 9n = 99  

-  (-27m) + 9n = -9

    54m + 0    = 108

54m = 108

54         54

m = 2

from equation (1),

27(2) + 9n = 99  

54 + 9n = 99  

9n = 99 - 54

9n = 45

9n = 45

9       9

n=5

Hence m=2 and n=5.