P(x)=3.P(-x)+4x-1
Find remainder when P(x-1) is divided by (x+2).
:
My work:
P(-3)=3.P(3)-13
there is no relation between P(x) and P(-x),middle term of both equations are different.One has a - sign,the other has not..
well.. one can always go the brute force route
$p(x) = 3p(-x) + 4x -1$
let $p(x) = c_0 + c_1 x + c_2 x^2 + c_3 x^3 \dots$
$p(-x) = c_0 - c_1 x + c_2 x^2 - c_3 x^3 \dots$
$c_0 = 3 c_0 -1 \Rightarrow c_0 = \dfrac 1 2$
$c_1 = -3c_1 + 4 \Rightarrow c_1 =1$
$c_2 = 3 c_2 \Rightarrow c_2 = 0$ and similarly for $c_k,~k>2$
$p(x) = x + \dfrac 1 2$
$p(x-1) = (x-1) + \dfrac 1 2 = x - \dfrac 1 2$
$\dfrac{x-\frac 1 2}{x+2} = -\dfrac 5 2$