First slide

Theory of equations

Question

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

Moderate

A

27
B

-27
C

5
D

-5

Solution

Since, $f (x) = x^{3} + 3 x^{2} - 9 x + λ = (x - α)^{2} (x - β)$
$∴ α$ is a double root.
$∴ f^{'} (x) = 0$ has also one root $α$ .
i.e. $3 x^{2} + 6 x - 9 = 0$ has one root $α$ .
$∴ x^{2} + 2 x - 3 = 0 or (x + 3) (x - 1) = 0$
has the root $α$ which can either -3 or 1.
If $α = 1, then f (1) = 0 gives λ - 5 = 0 \Rightarrow λ = 5$
If $α = - 3, then f (- 3) = 0$ gives
$\begin{array}{l} - 27 + 27 + 27 + λ = 0 \\ \Rightarrow λ = - 27 \end{array}$

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