First slide

Theory of equations

Question

If $x^{3} + 3 x^{2} - 9 x + c$ is of the form $(x - α)^{2} (x - β)$ ,then c is equal to

Moderate

A

27
B

-27
C

5
D

-5

Solution

$f (x) = x^{3} + 3 x^{2} - 9 x + c$ is of the form $(x - α)^{2} (x - β)$ , showing that a is a double root so that $f^{'} (x) = 0$ has also one root d,
i.e., $3 x^{2} + 6 x - 9 = 0$ has one root $α$ . Hence, $x^{2} + 2 x - 3 = 0$ or $(x + 3) (x - 1) = 0$ has the root $α$ which can be either -3 or 1.
If $α$ = 1, then f (x)=0 gives c- 5 = 0 or c=5.
If $α$ = -3, then f (x) = 0 gives
$\begin{aligned} - 27 + 27 + 27 + c = 0 \\ ∴ & c = - 27 \end{aligned}$

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