On dividing 2x2+3x+1 by a linear polynomial g(x), the quotient is 2x – 1 and remainder is ‘r’, where rϵR, then g(x) is
x + 2
f(x)=2x2+3x+1=(2x−1)g(x)+r, where g (x) is the divisor.
Let g(x)=(x−a)
When f(x) is divided by x−a, the remainder is f(a)
Thus, r=2a2+3a+1
Hence, 2x2+3x+1=(2x−1)(x−a)+2a2+3a+1
⇒2x2+3x+1=2x2−2ax−x+a+2a2+3a+1
⇒2x2+3x+1=2x2−(2a+1)x+2a2+4a+1
Comparing the coefficient of x on both sides, we get:
3=−(2a+1)
⇒4=−2a
⇒a=−2
Thus, g(x)=(x+2)