if x^2 + px + 1 is a factor of ax^3 + bx + c, then prove a^2 - c^2 = ab .

if x^2 + px + 1 is a factor of ax^3 + bx + c, then prove a^2 - c^2 = ab .


1 Answers

Aman Bansal
592 Points
11 years ago

Dear Sajin,

 Divide ax^3+bx+c by x^2+px+1. You will get remainder as

By Factor theorem, remainder must be zero in this case.

Thus, coefficient of x = 0 and constant term = 0.

Hence, b - a + ap^2 = 0 and c + ap = 0

From second eqn. we get p = -c/a. Now, putting this value in the first one, we get:

b - a + c^2/a = 0.

Multiplying both sides by a, we get ab - a^2 + c^2 = 0,

OR, a^2 - c^2 = ab

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Aman Bansal

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