If the polynomial x6 px5 qx4 x2 x 3 is divisible by x4 1 then the value of p2 q2 is ........

Quantitative Aptitude (CDS) Simplification

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If the polynomial x⁶+px⁵+qx⁴-x²-x-3 is divisible by (x⁴-1),then the value of p²+q² is

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Explanation:

Let f (x) = x6 + px5 + qx4 − x2 − x − 3
$\because$ f (x) is divisible by (x⁴ − 1 ).
Then, f (1)= f (−1)=0
Now, f (1) = 1 + p + q − 1 − 1 − 3 = 0 => p + q = 4 …(i)
and f (− 1) = 1 − p + q − 1 + 1 − 3 => q − p = 2 …(ii)
On solving Eqs. (i) and (ii), we get
q = 3 and p = 1
$\therefore$ p² + q² = 3²+ 1²= 10