First slide

Algebra of complex numbers

Question

If $x = 2 + 5 i$ then the value of $x^{3} - 5 x^{2} +$ $33 x - 19$ is equal to

Moderate

A

-5
B

-7
C

7
D

10

Solution

$x = 2 + 5 i \Rightarrow x - 2 = 5 i$
$\Rightarrow (x - 2)^{2} = (5 i)^{2} \Rightarrow x^{2} - 4 x + 4 = - 25$
$\Rightarrow x^{2} - 4 x + 29 = 0$ We now divide
$x^{3} - 5 x^{2} + 33 x - 19$ by $x^{2} - 4 x + 29$
$x - 1 x^{2} - 4 x + 29 x^{3} - 5 x^{2} + 33 x - 19$
$x^{3} - 4 x^{2} + 29 x - \oplus -_\begin{aligned} - x^{2} + 4 x - 19 \\ - x^{2} + 4 x - 29 \\ - \oplus - \end{aligned}___10$
Thus $\begin{array}{l} x^{3} - 5 x^{2} + 33 x - 19 = (x - 1) (x^{2} - 4 x + 29) + 10 = (x - 1) (0) + 10 = 10 \end{array}$

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