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Find the remainder when $x^3+ 3x^2 + 3x + 1$ is divided by $ x+\pi $
Given:
$x^3+ 3x^2 + 3x + 1$ is divided by $x+\pi$
To do:
We have to find the remainder when $x^3+ 3x^2 + 3x + 1$ is divided by $x+\pi$.
Solution:
The remainder theorem states that when a polynomial, $p(x)$ is divided by a linear polynomial, $x - a$ the remainder of that division will be equivalent to $p(a)$.
Let $f(x) = x^3+ 3x^2 + 3x + 1$ and $g(x) = x+\pi=x-(-\pi)$
So, the remainder will be $f(-\pi)$.
$f(-\pi) = (-\pi)^3+3(-\pi)^2+3(-\pi) + 1$
$= -pi^{3}+3\pi^{2}-3\pi+1$
Therefore, the remainder is $-pi^{3}+3\pi^{2}-3\pi+1$.
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