Question

Given x + 2 is a factor of the polynomial f(x) = 3x^3 + ax^2 + 4x + b and its remainder is -5 when divided by x-3 find values of a and b.

Answer 1

f(x) =3x^2+ax^2+4x+b
(x+2)is a factor of f(x)
Therefore, f(-2)=0
f(-2)=3*(-2)^3+a*(-2)^2+4*-2+b
= - 32+4a+b
f(x)=0
Or, - 32+4a+b=0
Or,4a+b=32......... (i)

f(x) when divided by (x-3) gives remainder - 5

f(3)=3*(3)^3+a*3^2+4*3+b
=81+9a+12+b
=93+9a+b


93+9a+b=-5
Or, - 98=9a+b........(ii)

Subtracting ii from i we get,
4a+b=32
- 9a-b=+98
_________
-5a +0=130
Or,a=130/-5
Or, a=-26

4a+b=32
Or, 4*-26 +b=32
Or, - 104-32=-b
Or, - 136=-b
Or, 136=b

ANSWER :a=-26, b=136