Polynomial of Degree4





In this page we are going to see how to solve polynomial of degree4. You can find an example problem with detailed step by step solution.

Question:

Solve the equation x⁴ + 4 x³ + 5 x² + 2 x - 2 = 0 which one root is -1 + √-1

Solution:

Let the given equation as f (x) = x⁴ + 4 x³ + 5 x² + 2 x - 2

This is the polynomial of variable x whose degree is 4. So we can say that this equation must have four roots. One of the root is given that is -1+i.Since it is a complex number we can consider that another root as -1- i

So let us take α = -1 + i  β = - 1 - i

By using these two roots we can find a quadratic equation which is the part of the original equation.First let us find the quadratic equation.

General form of any quadratic equation:

 x² - (α + β) x + α β = 0

α = -1 + i  β = - 1 - i

Sum of roots (α + β) = -1 + i - 1 - i

                            = - 2

Product of roots (α β) = (-1 + i) (- 1 - i)

                            = (-1)² - i²

                            = 1 - (-1)

                            = 1 + 1

                            = 2

Therefore the required quadratic equation 

                 x² - (-2) x + 2 = 0

                 x² + 2 x + 2 = 0

This is the part of the equation of polynomial of degree 4.

By solving this quadratic equation x² + 2 x - 1 = 0 we can get other two roots. We can not factorize this quadratic equation. For solving this quadratic equation we have to use formula.

a = 1   b = 2    c = -1

 x = [- b ± √(b² - 4 a c)]/2 a

 x = [- 2 ± √(2² - 4 (1) (-1))]/2 (1)

 x = [- 2 ± √(4 + 4)]/2

 x = [- 2 ± √8]/2

 x = [- 2 ± 2 √2]/2

 x = 2 [- 1 ± √2]/2

 x = - 1 ± √2

Therefore the four roots are -1 + i ,- 1 - i , - 1 + √2 , - 1 - √2.

polynomial of degree4 polynomial of degree4