Chapter 8: Q. 10 (page 556)

In the complex number system, solve the equation $3 x^{5} - 10 x^{4} + 21 x^{3} - 42 x^{2} + 36 x - 8 = 0$

Short Answer

Expert verified

The solution of the equation in the complex number system is $\{- 2 i, 2 i, 1, 2, \frac{1}{3}\} .$

Step by step solution

Step 1. Given Information

The given equation is $3 x^{5} - 10 x^{4} + 21 x^{3} - 42 x^{2} + 36 x - 8 = 0 .$

We have to solve the equation.

Step 2. Solve

$3 x^{5} - 10 x^{4} + 21 x^{3} - 42 x^{2} + 36 x - 8 = 0 (3 x^{5} + 21 x^{3} + 36 x) - (10 x^{4} + 42 x^{2} + 8) = 0 3 x (x^{4} + 7 x^{2} + 12) - 2 (5 x^{4} + 21 x^{2} + 4) = 0 3 x (x^{4} + 4 x^{2} + 3 x^{2} + 12) - 2 (5 x^{4} + 20 x^{2} + x^{2} + 4) = 0 3 x [x^{2} (x^{2} + 4) + 3 (x^{2} + 4)] - 2 [5 x^{2} (x^{2} + 4) + 1 (x^{2} + 4)] = 0$

Step 3. Solve

By proceeding with the calculation further

$3 x (x^{2} + 3) (x^{2} + 4) - 2 (5 x^{2} + 1) (x^{2} + 4) (x^{2} + 4) [3 x (x^{2} + 3) - 2 (5 x^{2} + 1)] = 0 (x^{2} + 4) (3 x^{3} + 9 x - 10 x^{2} - 2) = 0 (x^{2} + 4) (3 x^{3} - 3 x^{2} - 7 x^{2} + 9 x - 2) = 0 (x^{2} + 4) [3 x^{2} (x - 1) - (7 x^{2} - 9 x + 2)] = 0 (x^{2} + 4) [3 x^{2} (x - 1) - (7 x^{2} - 7 x - 2 x + 2)] = 0$

Step 4. Solve

By proceeding with the calculation further

$(x^{2} + 4) [3 x^{2} (x - 1) - (x - 1) (7 x - 2)] = 0 (x^{2} + 4) (x - 1) (3 x^{2} - (7 x - 2)) = 0 (x^{2} + 4) (x - 1) (3 x^{2} - 7 x + 2) = 0 (x^{2} + 4) (x - 1) (3 x^{2} - 6 x - x + 2) = 0 (x^{2} + 4) (x - 1) (x - 2) (3 x - 1) = 0$

Step 5. Writing the equation in the complex number system

As we know $(x^{2} + a^{2}) = (x + a i) (x - a i) and i^{2} = - 1 .$

Therefore,

$(x + 2 i) (x - 2 i) (x - 2) (3 x - 1) = 0$

Thus, the solution of the equation in the number system is $\{- 2 i, 2 i, 1, 2, \frac{1}{3}\} .$

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In the complex number system, solve the equation $3 x^{5} - 10 x^{4} + 21 x^{3} - 42 x^{2} + 36 x - 8 = 0$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Solve

Step 3. Solve

Step 4. Solve

Step 5. Writing the equation in the complex number system

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In the complex number system, solve the equation3x5-10x4+21x3-42x2+36x-8=0

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Solve

Step 3. Solve

Step 4. Solve

Step 5. Writing the equation in the complex number system

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Calculus

Probability and Statistics

Applied Mathematics

Mechanics Maths

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

In the complex number system, solve the equation $3 x^{5} - 10 x^{4} + 21 x^{3} - 42 x^{2} + 36 x - 8 = 0$