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 $α$ is a double root so that f'(x) = 0 has also one root $α$ , 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
$\begin{array}{l} - 27 + 27 + 27 + c = 0 \\ ∴ c = - 27 \end{array}$

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