Mathematics
Assertion (A) : If polynomial p(x) = x51 - 51 is divided by polynomial g(x) = x - 1, the remainder is 0.
Reason (R) : When a polynomial p(x) is divided by polynomial g(x - a), the remainder is p(a).
A is true, R is false.
A is false, R is true.
Both A and R are true.
Both A and R are false.
Factorisation
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Answer
By remainder theorem,
When a polynomial f(x) is divided by a linear polynomial (x - a), then the remainder is equal to f(a).
Given,
⇒ g(x) = x - 1
⇒ x - 1 = 0
⇒ x = 1.
Substituting value of x = 1 in p(x), we get :
⇒ p(x) = x51 - 51
⇒ p(1) = 151 - 51
⇒ p(1) = 1 - 51 = -50.
∴ Remainder = -50
∴ Assertion (A) is false.
Dividing polynomial p(x) by g(x - a), we get :
⇒ x - a = 0
⇒ x = a.
Substituting x = a in p(x), we get p(a).
∴ Remainder = p(a)
∴ Reason (R) is true.
Hence, Option 2 is the correct option.
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