1. Tardigrade
  2. Question
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  4. Let a polynomial P ( x ) when divided by x -1, x -2, x -3 leaves the remainder 4, 5, 6 respectively. If P(x) is divided by (x-1)(x-2)(x-3) and remainder is R(x), then find the value of R(100).

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Q. Let a polynomial $P ( x )$ when divided by $x -1, x -2, x -3$ leaves the remainder 4, 5, 6 respectively. If $P(x)$ is divided by $(x-1)(x-2)(x-3)$ and remainder is $R(x)$, then find the value of R(100).

Complex Numbers and Quadratic Equations

Solution:

Let $P(x)=(x-1)(x-2)(x-3) Q(x)+R(x)$, where $R(x)=\left(a x^2+b x+c\right)$
Now,$P(1)=R(1)=4 \Rightarrow a+b+c=4 $
$P(2)=R(2)=5 \Rightarrow 4 a+2 b+c=5 $ ....(i)
$P(3)=R(3)=6 \Rightarrow 9 a+3 b+c=6$....(ii)
$\therefore $ On solving, we get....(iii)
$a =0, b =1, c =3 \text {. }$
Hence, $R(x)=x+3 \Rightarrow R(100)=103$.