Question

Find the values of a and b. If division of polynomial P(x) = 9x³ -3x²+ax-b
with (x-1) leave remainder 2 and division of it by( x-2/3)
leaves
remainder -3​

Answer 1

Answer:

if g (x) = x-1

then, p (1)= 9×1^3 -3×1^2 +a×1 -b = 2

= 9-3+a-b=2

6+a-b=2

a-b = -4

a=-4+b................ (1)

b=4-a..................(2)

if g (x) = x-2/3

then, p (2/3) =9×(2/3)^3 -3×(2/3)^2 + a×2/3 - b= -3

=8/3 - 4/3 + 2a/3 - b = -3

=8-4+2a-3b/3= -3

= 4+2a-3b = -9

= 2a-3b = -13

= a = -13+3b/2.............. (3)

= b= 13-2a/3.................(4)

from (1) and (3),

-4+b = 13-3b/2

b = 13-3b/2 +4

b = 26-3b+8/2

2b = 34-3b

2b + 3b = 34

5b = 34

b = 34/5

from equation (1)

a = 4+b

now put the value of b in equation

a = 4+ 34/5

a = 20+ 34 /5

5a = 54

a = 54/5

hence, the value of a and b are 54/5 and 34/5 respectively.