Question : If the polynomial 36x4+48x3+27x2+21x+7 is divided by another polynomial 3x2+4x+1, the remainder comes out to be (2a+1)x-20b, find the value of a and b.
Doubt by Gauri
Solution :
p(x) = 36x4+48x3+27x2+21x+7
g(x) = 3x2+4x+1
g(x) = 3x2+4x+1
r(x) = (2a+1)x-20b
Dividing p(x) by g(x) by long division method.
____________________
3x2+4x+1)36x4+48x3+27x2+21x+7(12x2+5
36x4+48x3+12x2
- - -
------------------------------------------------
15x2+21x+7
15x2+20x+5
- - -
------------------------------------------------
x+2
------------------------------------------------
____________________
3x2+4x+1)36x4+48x3+27x2+21x+7(12x2+5
36x4+48x3+12x2
- - -
------------------------------------------------
15x2+21x+7
15x2+20x+5
- - -
------------------------------------------------
x+2
------------------------------------------------
r(x) = x+2
But r(x) = (2a+1)x-20b
Equating both of them
x+2=(2a+1)x-20b
equating the coefficient of x both sides
1=(2a+1)
1-1=2a
0=2a
a=0
x+2=(2a+1)x-20b
equating the coefficient of x both sides
1=(2a+1)
1-1=2a
0=2a
a=0
equating the constant term both sides
2=-20b
2/(-20)=b
-1/10 = b
b = -1/10
2=-20b
2/(-20)=b
-1/10 = b
b = -1/10