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Question

If (x2+px+1) is a factor of (ax3+bx+c), then


A

a2+c2=-ab

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B

a2-c2=-ab

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C

a2-c2=ab

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D

None of these

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Solution

The correct option is C

a2-c2=ab


Explanation for the correct option:

Applying basic concept of polynomial:

According to the given information, when we will divide (ax3+bx+c) by (x2+px+1), the remainder will be 0.

x2+px+1ax+apax3+bx+cax3+apx2+ax-apx2-ax+bx+capx2+ap2x+apb-a+ap2x+c+ap

⇒b-a+ap2x+c+ap=0⇒c+ap=0...(1)andb-a+ap2=0...(2)

From 1, we get

c+ap=0⇒p=-ca

By substituting the value of p in 2, we get

b-a+ac2a2=0⇒ab-a2+c2a=0⇒a2-c2=ab

Hence, option C is correct.


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