Question

If (x^3+ax^2+bx+6) has (x-2) as a factor and leaves a remainder 3 when divided by (x-3),find the values of a&b

Answer 1

Solution :- Let p(x) = x³ + ax² + bx +6

(x-2) is a factor of the polynomial x³ + ax² + b x +6

p(2) = 0 and also plug x= 2

p(2) = 2³ + a.2² + b.2 +6

=8+4a+2b+6

=14+ 4a+ 2b = 0

or 7 +2a +b = 0

b = - 7 -2a say equation (1)

x³ + ax² + bx +6 when divided by (x-3) leaves remainder 3.

p(3) = 3 and also plug x = 3

p(3) = 3³ + a.3² + b.3 +6

= 27+9a +3b +6

or 33+9a+3b = 3

or 11+3a +b =1 (divided by 3 )

3a+b = -10

or b= -10-3a�. say equation (2)

From (1) and( 2 ), we get

(- 7 -2a) = (-10 �- 3a)

-2a + 3a = -10 + 7

a = -3

Now , Substituting a = -3 in (1) equation , so we get

b = - 7 -2(-3)

= -7 + 6
= -1
or b = -1

So we got a = - 3 and b = -1 . That would be the final answer.