Question

Which of the following options hold?

Statement - I : If p(x) and g(x) are two polynomials with g(x) ≠ 0, then we can find polynomials g(x) and r(x) such that p(x) = g(x) x q(x) + r(x), where degree of r(x) is greater than degree of g(x).

Statement - II : When 4x⁵ + 3x³ + 2x² + 8 is divided by 4x² + 2x + 1, then degree of remainder is 1.

(A) Both Statement - I and Statement - II are true.

(B) Statement - I is true but Statement - II is false.

(C) Statement - I is false but Statement - II is true.

(D) Both Statement - I and Statement - II are false.

Answer 1

The correct option is: (C) Statement - I is false but Statement - II is true.

Explanation:

Statement - I is false because if p(x) and g(x) are two polynomials with g(x) ≠ 0, then we can find polynomials q(x) and (x) such that

p(x) = g(x) x g(x) + (x)

where (x) = 0 or degree of (x) < degree of g(x).

Statement - Il is true as when 4x⁵ + 3x³ + 2x² + 8 is divided by 4x² + 2x + 1, the remainder is - 5x/4 + 31/4 which is a polynomial of degree 1.