Question : Find what must be subtracted from the polynomial 2x4-4x3+4x2-4x+3 so that the obtained polynomial becomes divisible by x²+x+1.
Doubt by Bhavya
Solution :
Let
p(x)=2x4-4x3+4x2-4x+3
g(x)=x²+x+1
Now, let's find out what should be the remainder when p(x) is divided by g(x)
By Long Division Method
______________
x²+x+1)2x⁴-4x³+4x²-4x+3 (2x²+2
2x⁴+4x³+2x²
- - -
__________
2x²-4x+3
2x²+2x+2
- - -
___________
-2x+1
___________
r(x)=-2x+1
Hence, r(x) i.e. -2x+1 should be subtracted from p(x) so that the obtained polynomial becomes divisible by x²+x+1.
Doubt by Bhavya
Solution :
Let
p(x)=2x4-4x3+4x2-4x+3
g(x)=x²+x+1
Now, let's find out what should be the remainder when p(x) is divided by g(x)
By Long Division Method
______________
x²+x+1)2x⁴-4x³+4x²-4x+3 (2x²+2
2x⁴+4x³+2x²
- - -
__________
2x²-4x+3
2x²+2x+2
- - -
___________
-2x+1
___________
r(x)=-2x+1
Hence, r(x) i.e. -2x+1 should be subtracted from p(x) so that the obtained polynomial becomes divisible by x²+x+1.