#1**0 **

Let the remainder be ax + b. Then by the factor theorem, a = 4 and b = -1, so the remainder is 4x - 1.

Guest Dec 14, 2019

#3**+1 **

On division by (x - 1)(x - 2) we would have

\(\displaystyle \frac{f(x)}{(x-1)(x-2)}=q(x)+\frac{r(x)}{(x-1)(x-2)},\)

where q(x) is of degree two less than f(x) and r(x) is linear, ax + b say.

Multiply throughout by (x - 1)(x - 2) to get

\(\displaystyle f(x)=q(x)(x-1)(x-2)+ax + b\)

and now substitute x = 1 and x = 2.

That gets you

\(\displaystyle f(1)=2=a+b,\\ f(2)=3=2a +b.\)

Solve those simultaneously to get a = 1 and b = 1.

Guest Dec 15, 2019