The value of 2x4-5x3-7x2-x-41- when x-2-3i is-

The correct option is D 6 2x^4+5x^3+4x^2 x+41 —(1)when x = 2 √(3)i = (2+√(3)i) —(2) x^2 = (2+√(3)i)^2 = 4+3(i)^2+4√(3)i #160; =4 3+4 ...

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Byju's Answer

Standard X

Mathematics

Solving a Quadratic Equation by Completion of Squares Method

The value of ...

Question

The value of $2 x^{4} + 5 x^{3} + 7 x^{2} - x + 41$ , when $x = - 2 - \sqrt{3 i}$ is:

-4

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-6

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Solution

The correct option is D 6
$2 x^{4} + 5 x^{3} + 4 x^{2} - x + 41$ ---(1)
when $x = - 2 - \sqrt{3} i = - (2 + \sqrt{3} i)$ ---(2)
$x^{2} = (2 + \sqrt{3} i)^{2}$
$= 4 + 3 (i)^{2} + 4 \sqrt{3} i$
$= 4 - 3 + 4 \sqrt{3}!$ $(∵ i^{2} = - 1)$
$x^{2} = 1 + 4 \sqrt{3} i$ ---(3)
$x^{3} = x^{2} . x$
$= (1 + 4 \sqrt{3} i) [- (2 + \sqrt{3} i)]$
$= - (2 + \sqrt{3} i + 8 \sqrt{3} i + 12!^{2})$
$= - (2 - 12 + 9 \sqrt{3} i)$
$x^{3} = 10 - 9 \sqrt{3} i$ ---(4)
$x^{4} = (x^{2})^{2}$
$= (1 + 4 \sqrt{3} i)^{2}$
$= 1 + 48!^{2} + 8 i \sqrt{3}$
$= 1 - 48 + 8 i \sqrt{3}$ $(i^{2} = - 1)$
$x^{4} = - 47 + 8!^{3}$ ---(5)
From (1),(2),(3),(4) and (5)
$2 (- 47 + 8 i \sqrt{3}) + 5 (10 - 9 \sqrt{3} i) + 7 (1 + 4 \sqrt{3} i) - (- 2 \sqrt{3} i) + 41$ $- 94 + 16 i \sqrt{3} + 50 - 45 \sqrt{3} i + 7 + 28 \sqrt{3} i + 2 + \sqrt{3} i + 41$
$= 6$ d)

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Similar questions

Q. If

x = - 2 - \sqrt{- 3}

, find the value of

2 x^{4} + 5 x^{3} + 7 x^{2} - x + 41

Q. Evaluate the following:
(i)

2 x^{3} + 2 x^{2} - 7 x + 72, when x = \frac{3 - 5 i}{2}

(ii)

x^{4} - 4 x^{3} + 4 x^{2} + 8 x + 44, when x = 3 + 2 i

(iii)

x^{4} + 4 x^{3} + 6 x^{2} + 4 x + 9, when x = - 1 + i \sqrt{2}

(iv)

x^{6} + x^{4} + x^{2} + 1, when x = \frac{1 + i}{\sqrt{2}}

(v)

2 x^{4} + 5 x^{3} + 7 x^{2} - x + 41, when x = - 2 - \sqrt{3} i

Evaluate the following :

$(i) 2 x^{3} + 2 x^{2} - 7 x + 72, w h e n x = \frac{3 - 5 i}{2} (i i) x^{4} - 4 x^{3} + 4 x^{2} + 8 x + 44, w h e n x = 3 + 2 i (i i i) x^{4} + 4 x^{3} + 6 x^{2} + 4 x + 9, w h e n x = - 1 + i \sqrt{2} (i v) x^{6} + x^{4} + x^{2} + 1, w h e n x = \frac{1 + i}{\sqrt{2}} (v) 2 x^{4} + 5 x^{3} + 7 x^{2} - x + 41, w h e n x = - 2 - \sqrt{3} i$

Q. The polynomial

2 x^{4} - 5 x^{3} + 2 x^{2} - x + 2

is divisible by

Q. Find the value of a such that (x − 4) is a factors of 5x³ − 7x² − ax − 28.

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The value of 2x4+5x3+7x2−x+41, when x=−2−√3i is:

The value of $2 x^{4} + 5 x^{3} + 7 x^{2} - x + 41$ , when $x = - 2 - \sqrt{3 i}$ is: