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If p(x) is a common multiple of degree 6 of the polynomials f(x) = x3 + x2 – x – 1 and g(x) = x3 – x2 + x – 1, then which one of the following is correct ?

(a) p(x) = (x – 1)2 (x + 1)2 (x2 + 1) 

(b) p(x) = (x – 1) (x + 1) (x2 + 1)2 

(c) p(x) = (x – 1)3 (x + 1) (x2 + 1) 

(d) p(x) = (x – 1)2 (x4 + 1)

1 Answer

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Best answer

(a) p(x) = (x – 1)2 (x + 1)2 (x2 + 1) 

f(x) = x3 + x2 – x – 1 

g(x) = x3 – x2 + x – 1 

f(x) . g(x) = (x3 + x2 – x – 1) . (x3 – x2 + x – 1)

= x6 – x4 – x2 + 1

∴ p(x) = x6 – x4 – x2 + 1 

= x4 (x2 – 1) – (x2 – 1) = (x2 – 1) (x4 – 1) 

= (x – 1) (x + 1) [(x2)2 – 1] 

= (x – 1) (x + 1) [(x2 – 1) (x2 + 1)]

= (x – 1) (x + 1) [(x – 1) (x + 1) (x2 + 1)] 

= (x – 1)2 (x + 1)2 (x2 + 1).

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