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If a Polynomial $x^4+x^2+1$ is divisible by $x^2+m x+1$ and $x^2+n x+1$. Then $m+n$ is equal to
(1) 2
(2) 0
(3) 3
(4) 4
Options:
(1) 2
(2) 0
(3) 3
(4) 4
Solution:
2145 Upvotes
Verified Answer
The correct answer is:
0
$x^4+x^2+1$ is divisible by $x^2+n x+1$ and $x^2+m x+1$
$$
\therefore x^4+x^2+1=\left(x^2+m x+1\right)\left(x^2+n x+1\right)
$$
equating the coefficients of $x^3$ on both sides
$$
m+n=0
$$
$$
\therefore x^4+x^2+1=\left(x^2+m x+1\right)\left(x^2+n x+1\right)
$$
equating the coefficients of $x^3$ on both sides
$$
m+n=0
$$
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