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If \( \Large n^{2}px+1 \) is a factor of expression \( \Large ax^{3}+bx+c \) then:

A) \( \Large a^{2}+c^{2}=-ab \)
B) \( \Large a^{2}-c^{2}=-ab \)
C) \( \Large a^{2}-c^{2}=ab \)
D) none of these

Correct Answer:

A) \( \Large a^{2}+c^{2}=-ab \)

Description for Correct answer:

Given that \( \Large x^{2}+px+1 \) is factor of \( \Large ax^{3}+bx+c=0 \) then

let \( \Large ax^{3}+bx+c= \left(x^{2}+px+1\right) \left(ax+\lambda\right) \), where \( \Large \lambda \) is constant. Then equating the coefficient of like powers of x on both sides, we get.

\( \Large 0 = ap+\lambda, b=p\lambda+a, c=\lambda \)

\( \Large p=\frac{-\lambda}{a}=\frac{-c}{a} \)

Hence, \( \Large b= \left(-\frac{c}{a}\right)c+a \) or \( \Large ab=a^{2}-c^{2} \)

Part of solved Quadratic Equations questions and answers : >> Elementary Mathematics >> Quadratic Equations

Comments

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