Evaluate the following: x^4-4x^3+4x^2+8x+44, when x=3+2i - Vidyadip

Question

Evaluate the following:
$\text{x}^4-4\text{x}^3+4\text{x}^2+8\text{x}+44,$ when $\text{x}=3+2\text{i}$

✓

Answer

We have,
$\text{x}=3+2\text{i}$
$\Rightarrow\text{x}-3=2\text{i}$
$\Rightarrow(\text{x}-3)^2=(2\text{i})^2$
$\Rightarrow\text{x}^2+3^2-2\times3\times\text{x}=-4$
$\Rightarrow\text{x}^2+9-6\text{x}+4=0$
$\Rightarrow\text{x}^2-6\text{x}+13=0 \ ...\text{(i)}$
Now,
$\text{x}^4-4\text{x}^3+4\text{x}^2+8\text{x}+44$
$=\text{x}^2(\text{x}^2-6\text{x}+13)+6\text{x}^2-13\text{x}^2-4\text{x}^3+4\text{x}^2+8\text{x}+44$ (adding and subtracting 6x³ and 13x²)
$=\text{x}^2\times0+2\text{x}^3-9\text{x}^2+8\text{x}^2+44$ (using (i))
$2\text{x}(\text{x}^2-6\text{x}+13)+12\text{x}^2-26\text{x}-9\text{x}^2+8\text{x}+44$ (adding and subtracting 12x³ and 26x²)
$2\text{x}\times0+3\text{x}^2-18\text{x}+44$ (using (i))
$=3(\text{x}^2-6\text{x}+13)+5$
$=3\times0+5$ (using (i))
$=5$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "5 Marks Questions" from Complex Numbers→Complex Numbers — all question types→MATHS STD 11 Science question papers→Rajasthan Board STD 11 Science question papers→Rajasthan Board question paper generator→

Similar questions

Solve the following equations:
$\cot\text{x}+\tan\text{x}=2$

Prove that:
$\sin3\text{A}+\sin2\text{A}-\sin\text{A}=4\sin\text{A}\cos\frac{\text{A}}{2}\cos\frac{3\text{A}}{2}$

A bag contains 6 red, 4 white and 8 blue balls. If three balls are drawn at random, find the probability that:
i. one is red and two are white
ii. two are blue and one is red
iii. one is red.

Verify that the area of the triangle with verticies (2, 3), (-3, -1) remains invariant under the translation of axes when the origin is shifted to the point (-1, 3).

Differentiate the functions with respect to 'x'.
$\text{x}\cos\text{x}$

Prove that the centres of the three circles x²+ y² - 4x - 6y - 12 = 0, x² + y² + 2x + 4y - 10 = 0 and x² + y² - 10x - 16y - 1 = 0 are collinear.

Find the general solution for each of the following equations:
$\sin\text{x}+\sin3\text{x}+\sin5\text{x}=0$

Let a_n be the nth term of the G.P. of positive numbers. Let $\sum\limits_{\text{n}=1}^{100}\text{a}_{2\text{n}}=\alpha\text{ and}\sum\limits_{\text{n}=1}^{10}\text{a}_{2\text{n}-1}=\beta,$ such that $\alpha\neq\beta.$ Prove that the common ratio of the G.P. is $\frac{\alpha}{\beta}$

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow0}\frac{2\sin\text{x}^\circ-\sin2\text{x}^\circ}{\text{x}^3}$

Find the middle terms(s) in the expansion of:

$\Big(\text{x}-\frac{1}{\text{x}}\Big)^{2\text{n}+1}$