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Inequalities Involving Modulus Function

If x = 4+2i t...

Question

If x = 4+2i then prove that the value of the expression $x^{3} - 7 x^{2} + 12 x + 25$ is divisible by 1 and 5 , where (i= $\sqrt{- 1}$ )

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Solution

If x= 4+ 2i.
$f(x)=x^{3}-7x^{2}+12x+25$ ;
= $(4+2i)^{3}-7(4+2i)^{2}+12(4+2i)+25$ .----- (i)
Now,
$(4+2i)^{3}=64 + 96i-48-8i$ ;
$(4+2i)^{3}=16+88i$ ;
$(4+2i)^{2}=16-4+16i$ ;
Substituting this value in (i)...
$16+88i-7(12+16i)+12(4+2i)+25$ ;
$16+88i-84-112i+48+24i+25$ ;
So, $f(4+2i)=5$ ;
5 is divisible by 1 and 5.

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