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Question

Find the GCD of the following x3x2+x1,x41

A
(x1)(x2+1)
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B
(x+1)(x2+1)
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C
(x1)(x21)
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D
None of these
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Solution

The correct option is B (x1)(x2+1)
We first factorize the given polynomials x3x2+x1 and x41 as shown below:

x3x2+x1=x2(x1)+1(x1)=(x1)(x2+1)

x41=(x2)2(1)2=(x21)(x2+1)(a2b2=(a+b)(ab))=(x1)(x+1)(x2+1)

The common factors of x3x2+x1 and x41 are (x1) and (x2+1), therefore, the GCD is (x1)(x2+1).

Hence, the greatest common divisor is (x1)(x2+1).

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