If x3+ax3+10-bx is divisible by (x-2) . . .

Question : If x³+ax²+10-bx is divisible by (x-2) and (x-1) then find the values of a and b.

Doubt by Purvi

Solution :

p(x)=x³+ax²+10-bx

p(x)=x³+ax²-bx+10

Case I

g(x)=(x-2)
p(x) is divisible by g(x), it means g(x) must be a factor of p(x).

g(x)=0
x-2=0
x=2

p(2)=0
(2)³+a(2)²-b(2)+10=0
8+4a-2b+10=0
4a-2b+18=0
4a-2b=-18
2(2a-b)=-18
2a-b=-18/2
2a-b=-9 — (1)

Case II

g(x)=(x-1)
p(x) is divisible by g(x), it means g(x) must be a factor of p(x).

g(x)=0
x-1=0
x=1

p(1)=0
(1)³+a(1)²-b(1)+10=0
1+a-b+10=0
a-b+11=0
a-b=-11 — (2)

Subtracting equation (2) from equation (1)

2a-b-(a-b)=-9-(-11)
2a-b-a+b=-9+11
a+0=2
a=2
putting this value of a in equation (1)

2(2)-b=-9
4-b=-9
4+9=b
b=13

Hence, a=2 and b=13

Class 9