Question

Find the value of 2x^3-11x^2+44x+27if x=25/3-4i

Answer 1

Answer:  The required value of the given expression is 2.

Step-by-step explanation:  We are given to find the value of the following cubic expression if $x=\dfrac{25}{3-4i}:$

$E(x)=2x^3-11x^2+44x+27~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)$

We have

$x\\\\\\=\dfrac{25}{3-4i}\\\\\\=\dfrac{25(3+4i)}{(3-4i)(3+4i)}\\\\\\=\dfrac{25(3+4i)}{3^2-(4i)^2}\\\\\\=\dfrac{25(3+4i)}{9+16}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~[\textup{since }i^2=-1]\\\\\\=\dfrac{25(3+4i)}{25}\\\\=3+4i.$

Substituting the value of x in expression (i), we get

$E(3+4i)\\\\=2(3+4i)^3-11(3+4i)^2+44(3+4i)+27\\\\=2(27+108i+144i^2+64i^3)-11(9+24i+16i^2)+132+176i+27\\\\=54+216i-288-128i-99-264i+176+159+176i\\\\=2+0i\\\\=2.$

Thus, the required value of the given expression is 2.