Maths · Limits, continuity and differentiability
A polynomial \) when divided by leaves remainder Then
A polynomial \( p(x) \) when divided by \( x^{2}- \) \( 3 x+2 \) leaves remainder \( 2 x-3 . \) Then
- A. \( p(x) \) must have a root between 1 and 2
- B. \( p(x) \) cannot have a root between 0 and 3
- C. \( p(x) \) must have a real root but mayor may not be between o and 3
- D. \( p(x) \) need not have a real root
Step-by-step solution
Given p(x) divided by x^2-3x+2 leaves remainder 2x-3. Since x^2-3x+2 = (x-1)(x-2), we have p(x) = (x-1)(x-2)q(x) + 2x-3. Then p(1) = -1 and p(2) = 1. Since p is a polynomial, it is continuous, so by the Intermediate Value Theorem, there is a root in (1,2). Thus option A is correct.
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