First slide

Theory of equations

Question

If $x^{2} + px + 1$ is a factor of the expression ${ax}^{3} + bx + c,$ then

Moderate

A

$a^{2} + c^{2} = - ab$
B

$a^{2} - c^{2} = - ab$
C

$a^{2} - c^{2} = ab$
D

None of these

Solution

Given that $x^{2} + px + 1$ is factor of ${ax}^{3} + bx + c = 0,$ then let ${ax}^{3} + bx + c = (x^{2} + px + 1) (ax + λ)$ , where $λ$ is a constant. Then equating the coefficients of the powers of x on both sides, we get
$0 = ap + λ, b = pλ + a, c = λ$
$\Rightarrow p = - \frac{λ}{a} = - \frac{c}{a}$
Hence $b = (- \frac{c}{a}) c + a or ab = a^{2} - c^{2}$ .

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