First slide

Theory of equations

Question

The polynomial $x^{3} - 3 x^{2} - 9 x + C$ can be written in the form ${(x - a)}^{2} (x - b),$ then C =

Easy

A

2
B

3
C

25
D

27

Solution

The given equation is $f (x) = x^{3} - 3 x^{2} - 9 x + C$
Let $f^{|} (x) = 0$

$\Rightarrow 3 x^{2} - 6 x - 9 = 0$

$\Rightarrow x^{2} - 2 x - 3 = 0$

$\Rightarrow (x - 3) (x + 1) = 0$

$∴ x = 3, - 1$

$f (- 1) = {(- 1)}^{3} - 3 {(- 1)}^{2} - 9 (- 1) + C = 0 \Rightarrow C = - 5$ and

$f (3) = {(3)}^{3} - 3 {(3)}^{2} - 9 (3) + C = 0 \Rightarrow C = 27$

$∴ C = 27$

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